On the long time asymptotic behavior of the modified Korteweg de Vries equation with step-like initial data
Tamara Grava Alexader Minakov
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On the long time asymptotic behaviour  of the modified Korteweg de Vries equation  with step-like initial data
     Tamara Grava^† and   Alexander  Minakov^
^†SISSA, via Bonomea 265, 34136 Trieste, Italy and School of Mathematics, University of Bristol, UK
^Institut de Recherche en Mathématique et Physique (IRMP), Université Catholique de Louvain
(UCL), Louvain-la-Neuve, Belgium
grava@sissa.it      oleksandr.minakov@uclouvain.be

    
==================================================================================================================================================================================================================================================================================================================================


We study the long time asymptotic behaviour of  the  solution  q(x,t), x∈ℝ, t∈ℝ^+,  of the modified Korteweg de Vries equation  (MKdV)  q_t+6q^2q_x+q_xxx=0
  with step-like initial datum  

    q(x,0)→
    c_-  
    
    c_+  ,

with c_->c_+≥ 0.
For the  step 
 initial data

    q(x,0)=
    
    
    c_-  ,
    
    c_+  ,

the solution develops an oscillatory region  called dispersive shock wave region that   connects the two constant regions c_+ and c_-.
We show that the dispersive shock wave  is described by a modulated periodic travelling wave solution of the  MKdV  equation where the modulation parameters evolve
according to a Whitham modulation equation.



 The oscillatory region is expanding within a cone in the (x,t)  plane defined as 
-6c_-^2+12c_+^2<x/t<4c_-^2+2c_+^2, with t≫ 1.
  
For step-like initial data we show that the solution decomposes for long times in three main regions:

  *  a   region where solitons  and  breathers   travel with positive velocities on a constant background c_+;

  *  an expanding oscillatory region    (that generically  contains breathers);

  *  a region  of breathers travelling with negative velocities on the constant background c_-. 

When the oscillatory region does not contain  breathers, the form of the asymptotic  solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data.
The phase shift depends on  the solitons, breathers and the radiation of the initial data.
This shows that   the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons,  breathers and radiation.


§ INTRODUCTION

We consider the   Cauchy problem for the focusing modified Korteweg–de Vries  (MKdV) equation 

    q_t(x,t)+6q^2(x,t)q_x(x,t)+q_xxx(x,t)=0,   x∈ℝ, t∈ℝ^+,

with a step–like initial datum q_0(x)  of the form

    q(x,0)=
    q_0(x)→ c_±     as    x→±∞,

where c_± are some  real  constants.
We are interested in the long-time behavior of the solution. 


Due to symmetries q↦-q and x↦-x, t→-t, it is enough to consider the case

    c_-≥ |c_+|,
 since the other cases of mutual location of the constants c_-, c_+ can be reduced to it; however, the cases c_->c_+≥0 and c_->0>c_+≥-c_-  though qualitatively similar, leads to a quite different asymptotic analysis.
In the present manuscript   we restrict ourselves to the case c_->c_+≥0  and  we discuss  briefly the differences with respect to the case 
c_+<0,|c_+|≤ c_-  in the Appendix <ref>.

The   focusing MKdV equation  is a canonical model  for  the  description  of  nonlinear  long  waves  when there is a polarity symmetry, and it has many
physical applications;  in particular,  this includes waves in a quantized film <cit.>  internal ocean waves <cit.>, ion acoustic waves in a two component plasma <cit.>.

The MKdV equation  is an integrable equation  <cit.> with an infinite number of conserved quantities.
For the  class of initial data  considered, the classical mass and momentum   have to be replaced by the conserved quantities
  
    H_0 = ∫_-∞ ^ x (q(x, t)-c_-)%̣ṣx + ∫_x^+∞ (q(x, t) - c_+) %̣ṣx +(c_–c_+)x-2(c_-^3-c_+^3)t,
      
       
        H_1 = ∫_-∞ ^ x (q^2(x, t)-c^2_-)%̣ṣx + ∫_x^+∞ (q^2(x, t) - c^2_+) %̣ṣx +(c^2_–c^2_+)x-4(c_-^4-c_+^4)t.



  
























The study of the long-time asymptotic behaviour  of integrable  dispersive equations with   initial datum vanishing at infinity 
 was initiated in the mid-seventies  using the inverse scattering method in the works of  Ablowitz and Segur <cit.> and Manakov and Zakharov <cit.>. In the seminal paper <cit.> Deift and Zhou  
  introduced   the  steepest descent  method  for oscillatory  Riemann-Hilbert (RH) problems  to study the long-time asymptotic behaviour of the defocusing MKdV   equation with initial data vanishing at infinity. Such technique was extensively implemented in the asymptotic analysis of a wide variety of integrable problems  ( see e.g. <cit.>) (which in turn can be applied to some near-integrable cases, like the long-time behaviour of the perturbed defocusing nonlinear Schrödinger equation <cit.>).
  An extension of the  steepest descent  method  for oscillatory RH problems, called ∂̅ method, was introduced in <cit.>  and applied  to study the long-time behaviour of integrable dispersive equations with   initial data with low regularity  <cit.> in the strongly nonlinear regime.  In particular, for the MKdV equation  there is a vast body of literature studying   existence of solution
  for initial data with low regularity (see e.g. <cit.>). 
  Regarding the weakly nonlinear regime, the long-time asymptotic behaviour with small initial data   is quite similar for the focusing and defocusing MKdV equation and it was also  obtained without using  the integrability property 
   in <cit.>, <cit.> and  <cit.>.
  In the last years a vast literature of results concerns the long-time dynamics of initial boundary value problems of nonlinear dispersive equations. For a review see <cit.>.

  
  


The first results on the long time asymptotic analysis of Cauchy problems  with step like initial data were obtained for the Korteweg-de Vries (KdV) equation. Physicists have begun to understand the qualitative behaviour of the solution with the pioneering work of Gurevich and Pitaevsky <cit.>, who, working in the framework of Whitham theory <cit.>, predicted  the  appearance of high oscillations  called "dispersive shock waves".  These oscillations were described by modulated travelling waves.
This phenomenon was justified rigorously in the pioneering work of Khruslov <cit.> who, working in the framework of the inverse scattering theory, obtained formulas for the first finite number of peaks  of the oscillations (which were called  asymptotic solitons to distinguish them from  the usual solitons). This approach was extended to many other integrable models <cit.>.
Note that for a long time it was believed that Khruslov's solitons can be obtained from the Gurevich-Pitaevsky dispersive shock wave, until it was shown in <cit.> that dispersive shock waves (expressed in terms of elliptic functions) do not describe the asymptotic behaviour  of the wave in Khruslov's region, and the full matching of the two regions was obtained in <cit.>  for MKdV and in <cit.> for KdV.


Using ansatz for solutions of RH problems, Bikbaev <cit.> obtained interesting results for step-like quasi-periodic initial data.

 The KdV dispersive shock wave solution   emerges also in the small dispersion limit <cit.>. In particular, for the exact step initial data, the long time asymptotic and the small dispersion asymptotic description are equivalent, while for step like initial data the two asymptotic descriptions are   quite different. Indeed the dispersive shock wave obtained from  the long time asymptotic limit is always described  by the self-similar solution of the Whitham modulation equations <cit.>, while in the small dispersion limit  this is not generically the case  (see e.g. <cit.>, <cit.>).
 
 Implementation of the rigorous asymptotic analysis to step-like Cauchy problems for integrable equations started in the papers <cit.>.
 Since then,  the   long-time asymptotic behaviour of dispersive equations with  step-like  initial conditions has been studied for KdV in  <cit.>,  for the nonlinear Schrödinger equation in <cit.>, for the Camassa-Holm equation in <cit.>.
For the MKdV equation the analysis was initiated in the work <cit.> and  later  in <cit.> via the asymptotic analysis of the RH problem,  in <cit.> via matching ansatz method and in <cit.> via the Whitham method.  



The main feature in the long-time behaviour that distinguish  step-like initial conditions from decaying initial conditions is the formation of an oscillatory region that connects the  different behaviour at ±∞ of the solution.   These oscillatory regions are typically described by elliptic or hyperelliptic   modulated waves. 







































The scattering problem  for MKdV  with  non vanishing initial condition was developed in <cit.>, <cit.>, <cit.>.
The linear spectral problem  is a non self-adjoint problem  and for  a step-like  initial data  q_0(x) as in (<ref>) and satisfying certain assumption (see below) the Zakharov-Shabat or AKNS operator for (<ref>)
has a continuous spectrum r(k):Σ→ where Σ=ℝ∪[-ı c_-,ı c_-] and    generically  it might have   discrete spectrum anywhere in  ℂ\{ℝ∪ [-ı c_-,ı c_-]}, 
that corresponds to the zeros of a(k), the inverse of the transmission coefficient.
  Pure imaginary  couples of conjugated eigenvalues correspond to solitons, while quadruplets of complex conjugated eigenvalues correspond to breathers  <cit.>. A breather is a solution that is 
periodic in the time variable and decay exponentially in the space variable. 
 Unlike the KdV equation the MKdV equation can have higher order solitons and breathers.  In this manuscript we consider the  generic case when only first order solitons and breathers appear. Nongeneric cases are considered in the Appendix <ref>.
  First order solitons and breathers are    the  fundamental localised non radiating solutions   of the MKdV equation.
 
















Since the MKdV equation is not Galilean invariant,    solitary wave solutions  and breathers on a constant background c>0  cannot be  mapped to solutions on zero background. 

Our main result   contained in Theorem <ref> below,  is to show that the long-time asymptotic solution of the MKdV equation  with step-like initial data  of the form (<ref>) with c_->c_+≥ 0 decomposes into three main regions:

  *  a region of  solitons  and  breathers  on a constant background c_+  travelling in the positive direction;

  *  a  dispersive shock wave region, which connects the two different asymptotic behaviours of the initial data and interact elastically with breathers  and solitons.
 This  region is  described by a modulated travelling wave solution of  MKdV, or  by  a modulated travelling wave solution and breathers on an elliptic background.
 
  *  a  region  of breathers  on a constant background c_- travelling with a slower speed with respect to the dispersive shock wave. This region contains also radiation decaying in time.




The localised travelling wave   solution on a constant background  c>0 is parametrised by two  real  constants  ν and  κ_0>c, where  the points ±ıκ_0  constitute the  discrete spectrum of the Zakharov- Shabat
 linear operator, (namely they are the simple zeros of a(k))  and ν is the corresponding norming constant, and  takes the form






    q_soliton(x,t;c,κ_0,x_0)=c- 2 sign (ν)(κ_0^2-c^2)/κ_0cosh[2√(κ_0^2-c^2)x-(2c^2+4κ_0^2)t+x_0]-(ν) c,

  where  the phase shift x_0 depends on the spectral data via the relation 
    x_0=log2(κ_0^2-c^2)|ν|κ_0∈ℝ.
 The solution with  ν<0 is called soliton  and corresponds to  a positive hump, while the solution with ν>0  is called antisoliton and corresponds to a negative hump. In both cases  the speed  is 4κ_0^2 +2c^2, namely the speed of the soliton increases with the size of the step.
   The  maximal   amplitude of   the soliton is 2κ_0-c while the  minimal  amplitude  of the antisoliton is  -2κ_0-c.
This means that the values of the soliton span over the interval [0,2κ_0-c], with a pronounced peak at the maximal value 2κ_0-c, and the antisoliton ranges from 0 to -2κ_0-c, with a pronounced peak at the minimal value -2κ_0-c.



















The  breather solution  on a constant background c have been obtained using the bilinear method  and Darboux transformations  in <cit.>,  and inverse scattering in <cit.>.
In this manuscript we obtain the breather on a constant background as a solution of a RH  problem that is parametrised by the  complex number   κ,  with κ> 0, κ>0,   
 and by the complex parameter ν. 




Introducing the   complex number χ  defined as χ=χ_1+ıχ_2=√(κ^2+c^2), with χ_1>0, χ_2>0,  the breather solution  on a constant background takes the form

    q_breather(x,t;c,κ,ν)=c+
       +2∂_xarctan[|χ|cosφ +c|ν| χ_1^2/2|χ|^2 χ_2^- 2 Z χ_2|χ|^2/|ν|^2 Z χ_2+χ_1^2(|χ|^2-c^2)/4|χ|^2χ_2^2|ν|^ -2 Z χ_2+c sin(φ-θ_2)] ,
  where
 
    Z=x+4t(3χ_1^2-χ_2^2-3/2c^2),
         φ=2(Z-8t|χ|^2)χ_1+θ_1-θ_2,

 and  phases θ_1=-ν|ν| and θ_2=χ_1|χ|.
 
 Note that the denominator in (<ref>) can be written as cosh when |χ|>c and as sinh when |χ|<c. Despite having a sinh in the denominator, the expression remains regular  (see Remark <ref>  and cfr.  <cit.>).
 
 On the line Z=0    the breather oscillates with period π8|χ|^2χ_1
 and the envelope of the oscillations moves with  a speed 
 
    V=4χ_2^2+6c^2-12χ_1^2,     χ_1=√(κ^2+c^2),  χ_2=√(κ^2+c^2).

 We observe that for fixed κ and large values of c the velocity of the breathers is always negative.
 The level set  of the spectrum   in the complex  κ-plane corresponding to breathers with   equal speed is shown in Figure <ref>.
 




 It has been shown in <cit.> that solitons and breathers on a constant background  c>0  interact in an elastic way as in the case c=0.
 The breather (<ref>) turns into a pair of soliton and antisoliton  (<ref>) on a constant background when we let χ_1=0  and χ_2>0.

 

 
 



For c=0  we have χ_1=κ and χ_2=κ  and the  solution (<ref>) reduces to the standard  breather <cit.>
    q_breather(x,t;c=0,κ,ν)   =2∂_xarctan[(κ) cosφ(κ) coshΘ]
       =
    -4(κ)(κ) (κ) sinhΘcosφ+(κ) coshΘsinφ(κ)^2cos^2( φ)+(κ)^2cosh^2Θ ,

where 

    Θ=2κ(x+4(3(κ)^2-(κ)^2)t)+log2κ |κ|κ|ν|,

and
    φ(x,t)=2κ(x+4((κ)^2-3(κ)^2)t)+-ν|ν|-κ|κ|.

It has been shown in <cit.> that   the formation of breathers is generic  for certain 

compactly supported
 initial conditions.

 
The periodic travelling  wave solution of the MKdV equation  takes the form  (see Appendix<ref>)

    q_per(x,t;β_1,β_2,β_3,x_0   )=-β_1-β_2-β_3+
       +2(β_2+β_3)(β_1+β_3)/β_2+β_3-(β_2-β_1)cn^2√(β_3^2-β_1^2) (x-𝒱t)+x_0|m,


where β_3>β_2>β_1,  the speed 𝒱=2(β_1^2+β_2^2+β_3^2) and x_0 is an arbitrary phase. 
The function   cn(z|m) is the Jacobi elliptic function of modulus 


    m^2=β_2^2-β_1^2β_3^2-β_1^2,

and cn(z+2K(m)|m)=-cn(z|m) where K(m)=∫_0^π/2ds√(1-m^2sin^2s) is the  complete  elliptic integral of the first kind.
The periodic solution (<ref>) has wave number  k, frequency ω   and amplitude a  given by 

    k=π√(β_3^2-β_1^2)K(m),    ω=𝒱k,   a=2(β_2-β_1),

respectively.  When m→ 1,  the travelling wave solution (<ref>)  converges to the soliton solution  (<ref>)   with β_2=β_3=κ_0 and β_1=c.
 

  
 
 Our main result concerns the asymptotic description  for large times of the MKdV initial value problem for step-like initial data of the form (<ref>).
 Before stating our result, we  remark that the description   in <cit.>  of the long-time behaviour  of the solution of MKdV with  the  shock  initial data  
 
    q_0(x)=
    c_+     for  
    x>0,
    c_-     for    x≤0,

with c_->c_+>0  is as follows: there are two constant regions x/t<-6c_-^2+12c_+^2-δ and x/t>4c_-^2+2c_+^2+δ for any sufficiently small δ>0,   where the solution q(x,t)=c_∓+ o(1)   as t→∞  respectively. The   solution that connects the   two constant regions is oscillatory and it is  described   in terms of a  genus 2 quasi-periodic solution. 
In our  asymptotic analysis  we  show that such a genus 2 solution  is in fact  a genus 1 solution, and can be reduced to the modulated travelling wave solution  (<ref>) of MKdV, namely 
 
    q(x,t)=q_per(x,t,c_-,d,c_+,x_0)+O(t^-1),     -6c_-^2+12c_+^2+δ<x/t<4c_-^2+2c_+^2-δ,

    where   d=d(x,t) depends on   space and time  according to 
 
    xt=W_2(c_+,d,c_-).

 Here 
 
    W_2(β_1,β_2,β_3)=2(β_1^2+β^2_2+β_3^2)+4(β_2^2-β_1^2)(β_2^2-β_3^2)β_2^2-β_3^2+(β_3^2-β_1^2)E(m)/K(m),

 with β_1≤β_2≤β_3 and   E(m)=∫_0^π/2√(1-m^2sin^2θ)dθ  the complete elliptic integral of the second kind. We have  that
  
    W_2(c_+,c_+,c_-) =-6c_-^2+12c_+^2<x/t<  4c_-^2+2c_+^2=W_2(c_+,c_-,c_-).
  
  The quantity  W_2(β_1,β_2,β_3) is the speed  of the Whitham modulation equations <cit.> for β_1, β_2 and β_3   referring to   the Riemann invariants β_2. 
 The  solution  (<ref>)  is the  dispersive shock wave (see Figure <ref>)  and it was  first derived for  the Korteweg-de Vries equation <cit.>.  The Whitham equations for MKdV are strictly hyperbolic  and genuinely  nonlinear <cit.> only for β_2≠ 0 which implies that   the equation (<ref>)  can be inverted for d as a function of ξ only when c_+>0. The case c_->-c_+>0 can be studied in a similar way, by getting a slightly different evolution  of the wave  parameters (see the Appendix <ref> for further comments).  A comprehensive set of cases arising in the long-time asymptotic solution for the MKdV equation with step initial data  for various values of the parameters c_- and c_+ has been discussed in <cit.>.
 The phase x_0 of the travelling wave solution  (<ref>)  is given explicitly in terms of the   scattering data associated to the  MKdV equation.
 
We consider general step-like initial data and  we subject the initial function to the following conditions.
  
  The initial data  q_0(x) is assumed to be locally a function of bounded variation BV_loc() and  satisfying the following conditions



    ∫_-∞^+∞ |x|^2 |q̣_0(x)| <∞

and

    ∫_^±e^2 σ |x||q_0(x)-c_±|x̣<∞,

where σ>√(c_-^2-c_+^2)>0, and q̣_0(x) is the corresponding signed measure (distributional derivative of q_0(x)).
  This class includes the case of exact (discontinuous) step function (<ref>). Note that for a function q_0(x) which is locally of bounded variation and tends to c_± as x→±∞, the condition (<ref>) is equivalent to 

    ∫_-∞^0(1+|x|)|q_0(x)-c_-|x̣+∫_0^+∞(1+|x|)|q_0(x)-c_+|x̣<∞.

  Under Assumption <ref>, the initial value problem of the MKdV equation (<ref>) has a classical solution  for all t>0.
  
Under Assumption <ref>, the  inverse of the transmission coefficient a(k)  is analytic for  k∈ℂ_+ ∖[ı c_ł,0], and it has continuous limits to 
the boundary, 
with the exception of the points ı c_ł,ı c_, where a(k) may have at
most a fourth root singularity, namely a(k)√(k-ı c_±) is bounded  (see  Lemma <ref>). 
The zeros of a(k)

form the point spectrum and by analiticity, the number of zeros is finite.
   We fix the number of zeros in  the quarter plane k≥0 and k≥ 0  equal to N.
In order to formulate our results, we  enumerate the zeros   of a(k) in k≥0,k≥ 0 in the decreasing order of the speed of the corresponding solitons or breathers,
namely the points 

    κ_1,…,κ_N,  κ_j≥0,  κ_j≥ 0

correspond to the speeds

    +∞>V_1≥ V_2≥…≥ V_N>-∞.
 We recall that a soliton with the point spectrum κ on a constant background c has the speed 2c^2+4|κ|^2, while a breather with the point spectrum κ 
on a constant background c has the speed specified in (<ref>). The speed of a breather on a elliptic background is specified in (<ref>). In our case for each  breather  there are three options for large times:  it travels in  either the left constant background, right constant background, or dispersive shock wave background.
Note that it follows from properties 11  and 12 of Lemma <ref> below, that a(k) cannot have zeros  on (ı c_-,  0).  Therefore a soliton can travel only to the right of the dispersive shock vave.

We make further (generic) assumptions on the potential q_0(x) which are formulated in terms of  the  associated spectral function a(k).
The spectral function a(k) (see Definition <ref>) satisfies the following generic conditions:

  *  the zeros of a(k) are simple;

  *   the  zeros of a(k) do not lie on ℝ;
  *  all the speeds V_j of breathers and solitons are distinct,  namely  

    +∞>V_1> V_2> … > V_N>-∞;

  *  the behavior of a(k) at the points ı c_± is as follows: for some nonzero constants C_1, C_2+, C_2-,
    a(k)∼ C_1(k-ı c_-)^-1/4 k→ı c_-;
    
         
    a(k)∼ C_2,±(k-ı c_+)^-1/4 k→ı c_+±0,

where k→ı c_+±0 stands for the non tangential limit to the point ı c_+ from the left (+) and right (-)  side of the oriented segment [ı c_-,0], where the orientation is downward.


Similarly to Beals and Coifman <cit.>, one can show that the set of potentials satisfying Assumption <ref> form a dense open set in the space of all potentials satisfying Assumption <ref> with respect to  the  topology induced by (<ref>) but it is beyond the scope of this manuscript to prove this  here.











 The  question  we address  is: how,  under Assumptions <ref> and <ref>, 
the solitons, breathers  and  the dispersive shock wave interact for large values of time (see Figure <ref>).
    Theorem <ref>
  characterises these interactions in the general setting and (<ref>), (<ref>) and (<ref>)   below show explicitly how these interactions affect the asymptotic phase shifts of individual solitons  breathers and the dispersive shock wave.
  
  
 
Now   we  define

    T_j(k)=∏_l<j,κ_l>0k-κ_l/k-κ_lk+κ_l/k+κ_l·∏_l<j, κ_l=0k-κ_l/k-κ_l,   j≥ 2,     T_1(k)=1.

Let an initial datum satisfy 

Assumptions  <ref>, <ref> and 
 let δ>0 be a sufficiently small positive number such that all the breather/soliton speeds V_j satisfy |V_j-V_l|>4δ for all j≠ l,1≤ j,l≤ N.

Then the solution of the Cauchy problem for  the  MKdV   equation (<ref>)
  behaves for large t in the following way:

  
  (a) Soliton and breather region:  x/t>4c_-^2+2c_+^2+δ_1, δ_1>0.
  
For x/t such that |x/t-V_j|>δ for all j,
    q(x,t)=c_++𝒪(^-C t),
 with some C>0;

for |x/t-V_j|<δ for some j,
    q(x,t)={[ q_soliton(x,t; c_, κ_j, x_j) + 𝒪(^-C t),    κ_j=0,;                                                   ; q_breather(x,t; c_, κ_j,ν̂_j) +𝒪(^-C t),    κ_j>0, ].

where  q_soliton,   and q_breather are defined in (<ref>), (<ref>)  respectively and 
   
    ν̂_j = ν_j/T_j^2(κ_j),      
        x_j=ln(|κ_j|^2-c_+^2)T_j^2(ı|κ_j|)/|ν_j| |κ_j|
 and 
   
    T_j(k)=T_j(k)·exp[
       √(k^2+c_+^2)/2πı∫^ı c_+_-ı c_+lnT_j^2(s) ṣ/(s-k)(√(s^2+c_+^2))_+],

   with T_j(k) as in  (<ref>) and (√(s^2+c_+^2))_+ is the boundary value on the positive side of the oriented segment [ı c_+,-ı c_+].
    
  
  
  (b) Dispersive shock wave region: -6c_-^2+12c_+^2+δ_1<x/t<4c_-^2+2c_+^2-δ_1,  δ_1>0. 
  
  For x/t such that |x/t-V_j|>δ for all j,
  one has
  
    q(x,t)=q_per(x,t;c_-,d,c_+,x_0)+O(t^-1),

  where  the travelling wave  q_per has been defined in (<ref>),  the quantity  d=d(x,t)   depends on x and t as in (<ref>) and  the phase 
  
    x_0=-K(m)Δπ+K(m)

  depends on the discrete and continuous spectrum via the relation 

    Δ(x,t) =-√(c_-^2-c_+^2)K(m)[∫^ı c_ł_ı d(x,t)ln(|a(s)|^2T_n^2(s))sṣ/R_+(s)+
    ∫^ı c__0lnT_n^2(s)sṣ/R_+(s)]
       R_+(s)=√((s^2+c_+^2)(s^2+c_-^2)(s^2+d^2))_+
where the quantity T_n(s) is as in (<ref>), where n=n(x/t) is  the number of soliton/breather  speed  V_j  satisfying V_j>x/t,and a(k) is the  inverse of the  transmission coefficient associated to the  Zakharov-Shabat  spectral problem.
 For x/t such that |x/t-V_j|<δ for some j, one has  q(x,t)=q_be(x,t)+o(1) where q_be(x,t) is the breather solution on the  elliptic background and it  is specified by the solution of the RH  problem <ref> in Section <ref>. 
The  corresponding  speed  V_j  is given  in (<ref>).
  (c) Breathers on a constant background: x/t<-6c_-^2-δ_1
     or   -6c_-^2+δ_1 < x/t < -6c_-^2+12c_+^2-δ_1,δ_1>0. 

For t  large and  such that |x/t-V_j|>δ for all V_j,
    q(x,t)=c_-+𝒪(t^-1/2),

	and for x/t such that |x/t-V_j|<δ for some j,
    q(x,t)= q_breather(x, t; c_ł, κ_j, ν̂_j) + 𝒪(t^-1/2),

   where q_breather is defined in (<ref>) and the phase ν̂_j
    ν̂_j = ν_j
       
       /T_j^2(κ_j, ξ),

  with 
  
    T_j(k,ξ)=T_j(k) 
    exp   [-√(k^2+c_ł^2)/2πı{∫_ı c_ł^ı d_0(ξ)-∫^-ı c_ł_-ı d_0(ξ)ln|a(s)|^2ṣ/(s-k)(√(s^2+c_ł^2))_++..
       +..
    ∫_ı c_-^-ı c_-lnT_j^2(s)ṣ/(s-k)(√(s^2+c_ł^2))_+}]

for -c_ł^2/2<ξ<-c_ł^2/2+c_^2 with  d_0(ξ)=ı√(ξ+c_-^2/2), 
and
 
    T_j(k,ξ)   =  T_j(k) 
    exp[-√(k^2+c_ł^2)/2πı{∫_ı c_ł^0-∫^-ı c_ł_0ln|a(s)|^2ṣ/(s-k) (√(s^2+c_ł^2))_+..
       ..+∫_ı c_-^-ı c_-lnT_j^2(s)ṣ/(s-k)(√(s^2+c_ł^2))_+-∫_-k_0(ξ)^k_0(ξ)ln1+|r(s)|^2ṣ/(s-k)√(s^2+c_ł^2)}],

 for ξ<-c_-^2/2 and k_0(ξ)=√(-ξ-c_-^2/2)  and T_j(k) as in (<ref>).

 






















     
The  subleading term  of order 𝒪(t^-1/2) of the expansion of q(x,t) as t→∞  in the   left constant region is oscillatory and is described by the theorem below.

Away from breathers, the subleading term of the expansion of q(x,t) as t→∞  in the regions
x<(-6c_-^2-δ_1)t and (-6c_-^2+δ_1)t<x<(-6c_-^2+12c_+^2-δ_1)t
is  given by the formula 


    q(x,t)   =c_-+
    √(|ν(ξ)|√(-ξ+c_-^2/2)/3t |ξ+c_-^2/2|)×
       ×cos[
    16t-ξ+c_-^2/2^3/2
    +ν(ξ)ln192  t  (ξ+c_-^2/2)^2/√(-ξ+c_-^2/2)
    +
    ϕ(ξ)
    ]
    +𝒪(t^-1),

where ξ=x/12t and the phase shift and parameter ν(ξ)  are different for different regions of ξ=x/12t:
∙ for ξ<-c_-^2/2 they are given by the formulas

    ν(ξ)=1/2πln(1+|r(k_0)|^2),   k_0=√(-ξ-c_-^2/2), 
       ϕ(ξ):=π/4- r(k_0)-Γ(ıν(ξ))+χ^2(k_0),

and χ(k_0)=lim_k→ k_0T(k)k-k_0/k+k_0^ıν(ξ),
∙ and for -c_-^2/2<ξ<-c_-^2/2+c_+^2 they are given by the formulas

    ν(ξ)=1/2πln(1-|r_+(ı d_0)|^2)<0,
      ı d_0 = ı√(ξ+c_-^2/2),
       ϕ(ξ)=-π/4- r_+(ı d_0)-Γ(ıν(ξ))+χ^2(ı d_0),

and χ(ı d_0)=lim_k→ı d_0+0T(k)k-ı d_0/-(k+ı d_0)^ıν(ξ),|χ(ı d_0)|=1.Note that the term π/4 in the phase shifts has different signs for different values of ξ, and that the parameter ν(ξ) has different signs in different regions.
We also  observe  that the amplitude in the formula (<ref>) does not blow up when ξ approaches -c_-^2/2 if c_+>0, since  ν(ξ) has a first order zero as ξ=-c_-^2/2  for c_+>0 while it blows up at c_+=0. This suggests the existence of a non trivial transition zone.


  Our analysis is obtained by formulating the inverse scattering problem for the MKdV equation with step initial data as a RH  problem and then we implement 
   the long-time asymptotic analysis via the Deift-Zhou steepest descent method <cit.>. The proof that the oscillations in the  oscillatory region that were obtained in <cit.> via genus two theta functions, can be reduced to the dispersive shock wave solution for the MKdV equation, is obtained by “folding" the genus two Riemann Hilbert problem to a genus one RH  problem with poles. Further  analysis permits us to get rid of the poles and  to reduce the solution to the travelling wave solution of the MKdV equation.
   The estimate of the errors and the calculation of the subleading terms of the expansion   in Theorems <ref> and Theorem <ref> is obtained via the construction of local parametrices in terms of 
Airy functions and parabolic cylinder functions.

      We illustrate our results with the following examples.
   
   For the exact step (<ref>) one has   that the  reflection coefficient and the inverse of the transmission coefficients are  <cit.>
    r(k)=γ(k)^2-1γ(k)^2+1,   
    a(k)=1/2(γ(k)+1/γ(k)), γ(k)=((k-ı c_-)(k+ı c_+)(k+ı c_-)(k-ı c_+))^1/4 .

The dispersive shock wave is given by the  relation (<ref>)  with phase shift

    x_0=-√(c_-^2-c_+^2)π∫^ı c_ł_ı d(x,t)ln(|a(s)|^2)sṣ/√((s^2+c_+^2)(s^2+c_-^2)(s^2+d^2))+K(m) ,

The evolution for such initial data is illustrated in Figure <ref>. The  leading edge of the oscillatory region has been studied in 
 <cit.>, where it has been identified with a train of asymptotic solitons, 
and it has been shown that the amplitude of the first soliton   is approximately described

    q(x,t)≃ q_soliton(x,t;c_+,c_-,x̃_0)

where x̃_0=3/2log t+α for some constant α.  Here q_soliton(x,t;c_+,c_-,x̃_0) is the soliton solution  (<ref>)
 on the constant background c_+ with spectrum c_-. The highest peak of the first soliton is  approximately 
 located at  the position x_s(t)=2(c_+^2+2c_-^2)t-1/23/2log t+α/√(c_-^2-c_+^2).
 
The transition region between the dispersive shock wave and asymptotic solitons turned out to be very rich, and has been studied recently in <cit.>.

 
 For an initial function in the form of a soliton on a constant background (<ref>) on the left, and a constant on the right, i.e. 

    q_0(x) = 
    
    q_soliton(x,0;c_ł, κ_0,x_0),   x<0,      κ_0>c_ł>0, 
    
    
    c_,   x>0,   c_∈ℝ.

Here x_0=log2(κ_0^2-c_-^2)|ν|κ_0∈ℝ  and ν≠ 0.

 The   evolution is shown in Figure <ref>, where it is shown that the initial  approximate "soliton" decomposes in two solitons that pass rapidly through the dispersive shock wave.
 
 Consider the initial function in the form of a  constant c>0 on the left, and a soliton on  a zero background on  the right, i.e. 
    q_0(x)=c>0,   x<0, 
    -2κ_0 sgn(ν)/cosh[2κ_0(x-x_0)],    x>0,  ν∈ℝ∖{0},  c>κ_0>0,    x_0=1/2κ_0ln|ν/2κ_0|.

This initial data does not satisfy the decay at infinity of Assumption <ref>. Below we  show that the reflection coefficient is singular on the imaginary axis [ı c,0].
The reflection coefficient  is  r(k)=b(k)/a(k)  with 

    a(k) =1/2[γ(k)+γ(k)^-11-ıα/k+ıκ_0-γ(k)-γ(k)^-1ıβ/k+ıκ_0],
    
       b(k) = 1/2[γ(k)-γ(k)^-11+ıα/k-ıκ_0-γ(k)+γ(k)^-1ıβ/k-ıκ_0],

where 

    γ(k)=√(k-ı c/k+ı c),  α = 2ν^2κ_0/4κ_0^2+ν^2,
      β = 4νκ_0^2/4κ_0^2+ν^2.

The coefficient  b(k) has  a pole at k=ıκ_0 and the  function a(k) has two zeros k_1, k_2 in the half-plane  k≥0 :

    k_1,2 = ±β√(c^2+2cβ-(α-κ_0)^2)
    +ı(α-κ_0)(β+c)c+2β

and they are symmetric w.r.t. imaginary axis: k_1=-k_2.
Let us notice that when x_0>0 is not small, and hence ν>0 is big, we have α∼2κ_0,β∼ 0, and hence 

    k_1,2∼±0+ıκ_0.


Therefore the soliton  part of the initial data is  a breather from the spectral point of view,  with point spectrum very close to the imaginary axis.
The velocity of the breather is approximately 4κ_0^2<4c^2   while the velocity of the dispersive shock wave is  -6c^2<x/t<4c^2.
Namely the breather has a  positive velocity that is smaller then the velocity of the leading front of the dispersive shock wave and it will remain trapped by the dispersive shock wave. Furthermore,   since b(k) has a pole at k=ıκ_0, also  the reflection coefficient has a pole at k=ıκ_0, and this pole requires a quite delicate asymptotic analysis
that is beyond the scope of the present article.
In the physical literature <cit.> this phenomenon received the  name of `soliton trapping' inside the dispersive shock wave  (see Figure <ref>).




















This manuscript is organized as follow. 
We study the direct scattering problem and the main properties of the scattering data and the solvability of the associated RH  problem in section 2.
In section 3 we formulate  and solve  the RH   problems associated to the model problems that will be used in asymptotic analysis,
namely the  RH  problem for a breather and a soliton on a constant background and  then the model problems for the periodic 
 solution that is expressed via  hyperelliptic curves. We will then show how to reduce the  hyperelliptic solution to a travelling wave solution of MKdV.

In section 4 we start the asymptotic analysis introducing the g-function and performing the contour deformation and asymptotic analysis 
in the elliptic region and in section 5 we perform the contour deformation   and asymptotic analysis in the left and right constant regions,
to arrive at our main result, namely the 
proof of theorem <ref>.  In section 6 we calculate the subleading order correction of the left constant region thus proving  theorem <ref>.
Several appendices are used  to prove the most technical results.

  
 
  


































































































































§ PRELIMINARIES


Let us briefly describe what will happen in this Section.

In section <ref> we review the Lax pair formulation of the MKdV equation and we construct the Jost solutions at time t=0.
In section <ref>  we analyze the direct scattering problem following and  extending the derivation in <cit.> for our class of initial data.
In Section <ref>
 we introduce the RH problem <ref>, where we imposed a time dependence of the  form e^i4k^3 t. 
The way we arrive to the RH problem <ref> is heuristic, nevertheless, after the RH problem <ref> is stated, it serves as a basis for the rest of the section.
 
In Section <ref> we follow <cit.> and, first, we prove that the solution of the RH problem exists and admits differentiation with respect to the parameters x,t, and, second, that the columns of the solution of the RH problem <ref> are solutions of the overdetermined system which constitutes the Lax pair (<ref>), (<ref>). This allows to define a function q(x,t), which satisfies the MKdV equation (<ref>) and equals to the initial datum at t=0.

Note that the existence of RH problem's solution is guaranteed a priori by the Zhou's Schwartz reflection argument <cit.>, but in order to guarantee the differentiability in the parameters x,t we need either to require analytic continuation of the reflection coefficient off the real axis, or to require sufficiently fast decay of the reflection coefficient at the infinite parts of the RH problem's contour. The former can be achieved by imposing the exponential decay (<ref>), and  the latter by requiring sufficient smoothness of the initial function. 
Neither of these function spaces is embedded in each other, but we chose to impose the exponential decay of the initial datum  rather than the smoothness condition. This allows us 
 to treat  the  important cases of discontinuous initial functions described in Examples (<ref>)-(<ref>).



Results similar to that of Theorem <ref> but written for large x→-∞ and  fixed t allow to conclude that |q(x,t)|∼|r(-x/12t)|/√(x)∼|x|^-j/2-3/4 for an initial datum   q(x,0) whose j-th derivative is locally a function of bounded variation.  In the above the symbol ∼ means of the same order. Thus, the function q(x,t) at a later time t>0 satisfies the first moment condition (<ref>) only when j≥ 3, which is a much stronger  condition than the  conditions set in Assumption <ref>.
Note that the classical direct analysis of the x-equation (<ref>) allows to guarantee the existence of the Jost solutions only under the convergence of the first moment (<ref>).


 §.§ Lax pair and  Jost solutions

The MKdV equation (<ref>) admits the Lax pair representation in the form 

of an over-determined system of linear ordinary equations for a 2×2 matrix-valued function Φ(x,t;k), where k∈ is the spectral variable
<cit.>, 

    Φ_x(x,t;k)=(-ı kσ_3+Q(x,t))Φ(x,t;k),
          Φ_t(x,t;k)=(-4ı k^3σ_3+Q(x,t;k))Φ(x,t;k),

where Φ_x and Φ_t stand for partial derivative with respect to x and t respectively, σ_3=[  1  0;  0 -1 ] and 
    Q(x,t)=[       0  q(x,t); -q(x,t)       0 ],  Q(x,t;k)=4k^2Q-2ı k(Q^2+Q_x)σ_3+2Q^3-Q_xx.
 The compatibility condition Φ_xt=Φ_tx is equivalent to the MKdV equation. 
If we substitute q(x,t)=c (constant)  in (<ref>), then the Lax pair equations (<ref>), (<ref>) admit  explicit solutions  <cit.>
    E_0(x,t;k)=^-(i k x+4ik^3t)σ_3,

for c=0 and 

    E_c(x,t;k)   =12[ γ(k)+γ^-1(k) γ(k)-γ^-1(k); γ(k)-γ^-1(k) γ(k)+γ^-1(k) ]^-g_c(x,t;k)σ_3,   
       γ(k)=√(k-ı c/k+ı c),
         g_c(x,t;k)=ı( x +2t(2k^2-c^2)√(k^2+c^2),

 for  c≠ 0; the root functions  are defined in such a way that they have branch cut across the interval [ı c,-ı c].  In particular
 √(k^2+c^2) is real for k∈∪[ı c,-ı c]  and positive for k= 0+ where 0+ means the non tangential  limit from the  right with respect to the imaginary axis. 
Here and in the rest of the manuscript  the imaginary axis is oriented downwards and for this reason we use the notation [ı c,-ı c]  for the oriented segment with endpoints ±ı c_-.


 §.§ Direct scattering

 Below we give a quick review of the direct scattering problem for step-like initial data  q(x,0)(<ref>).  
 
 
 
 
  We denote E^(x;k)≡ E_c_(x,0;k)   and E^ł(x;k)≡ E_c_ł(x,0;k)  where E_c_±(x,t;k) is the matrix defined (<ref>).
Now we   consider equation (<ref>) for t=0 and  define the  Jost solutionsΦ^±(x;k):=Φ^±(x,0;k),  which satisfy the equation (<ref>), and have the (defining) property that 

    Φ^±(x;k)=E^±(x;k)(1+o(1))  ,   k∈ℝ∪[ı c_±, -ı c_±].

The Jost solutions  have the following integral representation

    Φ^±(x;k)=E^±(x;k)+∫^x_±∞L^±(x,y)E^±(y;k)dy,

where the kernels L^±(x,y) are independent from k and they are studied  in the Appendix <ref>.
From the above expression  it is  clear that the Jost functions are continuous  in the variable k for k∈ℝ∪(ı c_±, -ı c_±).
The following two Lemmas appeared before in <cit.>, <cit.>  under the condition c_+=0 or  c_+>0 for  the exact step,  ( see also <cit.>) but their proof was skipped there.

 Suppose the the initial data q(x,0) satisfies Assumption <ref>  and Assumption <ref>. Then the  Jost solutions  (<ref>)  their columns Φ^±_j(x;k),
 j=1,2 and their entries Φ^±_ij(x;k) have the following
properties.

  * Φ^±(x;k)=1.
  
  * Φ_1^+(x;k) is analytic in k∈C^-∖[0,-ı c_],  and continuous  up to the boundary
Φ_2^+(x;k) is analytic in k∈C^+∖[ı c_,0]  and continuous  up to the boundary,
Φ_1^-(x;k) is analytic in k∈C^+∖[ı c_ł,0] and continuous   up to the boundary
Φ_2^-(x;k) is analytic in k∈C^-∖[0,-ı c_ł] and continuous   up to the boundary.
    
   here ℂ^± = {k: ± k>0};
  *     Let  Φ(x;k) denote  either Φ^ł(x;k) or Φ^(x;k). By the reality of q(x) we have the symmetries:
  
    Φ_1(x;k)     =[  0  1; -1  0 ]Φ_2(x;k),
    Φ_1(x;-k)   =[  0  1; -1  0 ]Φ_2(x; k),
    Φ(x;-k)   = Φ(x,k).

  *  Large k asymptotics:
   
    [ Φ^_ 1(x;k)e^+ikx,  Φ^ł_2(x;k)e^-ikx ]
       =I+O(1/k),   k→∞,    k≤ 0,

    [ Φ^ł_1(x;k)e^+ikx,  Φ^_ 2(x;k)e^-ikx ]
      =I+O(1/k),   k→∞,    k≥0,

 where I stands for the 2× 2 identity matrix.
  
  *  Jump conditions:  
Φ_-(x;k)=Φ_+(x;k)(
  [ 0 i; i 0 ]),  
  k∈( c,- c),

  where Φ(x;k) and c denote Φ^ł(x;k) and c_ł
  or Φ^(x;k) and c_, respectively,
  and Φ_±(x; k)      denotes the non-tangential limits of the matrix
  Φ(x;k)  from the left (+) and from the right (-) of the  segment ( c, - c)  which is oriented from ı c to -ı c (i.e., Φ_±(x;k) = lim_ϵ→+0Φ(x;k±ϵ)).
The full proof of the Lemma is 

given in the Appendix C.
Here we will only prove properties 1,  3  and 5.
 Property 1. 
The functions Φ^± satisfy the x-equation (<ref>) for t=0; since the trace of the function -ı k σ_3+Q equals 0, by Liouville's formula for determinants, it follows that   Φ^±(x;k) are independent of x. From the large x asymptotics   (<ref>) we then obtain that the determinants are equal to 1.
 Property 3.
Let us write the  x-equation (<ref>) element-wise,

    Φ_11,x(x;k) + ı k Φ_11(x;k) = q(x) Φ_21(x;k),
    
    Φ_21,x(x;k) - ı k Φ_21(x;k) = -q(x) Φ_11(x;k),
    
       Φ_12,x(x;k) + ı k Φ_12(x;k) = q(x) Φ_22(x;k),
    
    Φ_22,x(x;k) - ı k Φ_22(x;k) = -q(x) Φ_12(x;k).

Changing in the last two equations k→- k and using the large x-asymptotics of Φ_ij, i,j=1,2, we get that Φ_12(x;-k)=-Φ_21(x;k),  Φ_11(x;k)=Φ_22(x;-k) and Φ_ij(x;-k) = Φ_ij(x;k),  i,j=1,2.

Indeed, the two pairs of vector-valued functions,
(Φ_22(x;-k), -Φ_12(x;-k)) and 
(Φ_11(x;- k),  Φ_21(x;- k)) satisfy the same systems of equations as (Φ_11(x;k), Φ_21(x;k)) and the same large x asymptotics, and hence coincide with the latter.
 Property 5. We use the defining property (<ref>) of the functions Φ^± and note that the limiting values of E^± on the two  sides of the  oriented  interval (ı c_±,-ı c_±) satisfy
E^±_-(x;k)=E^±_+(x;k)[ 0 ı; ı 0 ].
Thus, both functions Φ^±_-(x;k) and Φ^±_+(x;k+0)[ 0 ı; ı 0 ] satisfy the same differential equation (<ref>) (at t=0), and the same asymptotics as x→-∞, and thus they are equal.










Since the matrix-valued func­tions Φ^±(x ;k) 
 are solutions of the first order
dif­fer­en­tial equation (<ref>)
, they are
linearly dependent, i.e. there exists an x-independent  transition matrixT(k), such that


    T(k)=(Φ^(x;k))^-1Φ^ł(x;k),  
    k∈ℝ∪(ı c_,-ı c_).

Due to symmetries of the x-equation (<ref>) the transition matrix has the following structure:

    T(k)=[  a(k) -b(k);  b(k)  a(k) ],     a(k)=Φ^ł_1,Φ^_ 2, b(k)=Φ^_1,Φ^ł_ 1.
Note that the representation (<ref>)  and Lemma <ref>  allow to extend the domains of definition of the spectral functions a,b: the  function a(k) is analytic for k∈ℂ∖[ı c_-,0], and is defined also on both sides of the interval (0,-ı c_), and b(k) is defined for k∈ℝ∪(ı c_+, -ı c_-).
 By property 1  of Lemma <ref>, we have that T(k)=1 so that

    a(k) a(k)+b(k)b(k)=1.

The functions a^-1(k) and 
    r(k):=b(k)/a(k),   k∈ℝ∪(ı c_+,-ı c_+)
 

are called the (right)  transmission and  reflection coefficients, respectively.
Below we sumarize the analytical properties of the functions a(k), b(k) and r(k).  
We denote by 
a_±(k), b_±(k) and r_±(k) the non-tangential limits of  a(k), b(k) and r(k) 
   from the left (+) and from the right (-) of the  segment ( c_-, - c_-)  which is oriented from ı c_- to -ı c_-.
Suppose that  the initial data q(x,0) satisfies Assumption <ref>  and Assumption <ref>. 
Then the spectral functions a(k), b(k), r(k) have the following properties enumerated below.

  *  Analyticity: 
a(k) is analytic for  k∈ℂ^+ ∖[ı c_ł,0], and it can be extended continuously up to the boundary, 
with the exception of the points ı c_ł,ı c_, where a(k) may have at
most a fourth root singularity (k-ı c_±)^-1/4(i.e. the function a(k)(k-ı c_±)^1/4 is bounded as k→ı c_±).
The function a(k) might have at most a finite number of zeros for  k∈ℂ_+ ∖[ı c_ł,0].

The function b(k)   is defined for  k∈ℝ∪(ı c_,-ı c_+).

The function r(k) is defined for  k∈ℝ∪(ı c_,-ı
c_) except for the points where a(k)=0.


 Under the condtion (<ref>),
 the function a(k)  and b(k) have an analytic extension  in the δ-neighbor­hood of Σ=ℝ∪[ı c_ł,-ı c_ł], where δ= √(σ^2+c_+^2)-c_- and σ>√(c_-^2-c_+^2)>0, with σ as in Assumptions <ref>. The function r(k) is meromorphic in the same domain  with poles at the zeros of a(k).
  *  Asymptotics:

    a(k)=1+𝒪(k^-1)    as    k→∞,  k≥ 0;
    
       r(k)=𝒪(k^-1).

  *  Symmetries: in their domains of  definition 
    a-k=a(k),  b-k=b(k),   r-k=r(k).

  *  On
 (ı c_,-ı c_):

    a_-(k)=a_+k,   b_-(k)=-b_+k,   r_-(k)=-r_+k,

and on (ı c_ł,ı c_):
    a_-(k)=-ı b_+k,   a_+(k)=ı b_-k.

  *  Let 
    f(k):=i/a_-(k)a_+(k),   k∈ (ı
    c_ł,0),

then  on (ı c_ł,ı c_):
    f(k)=ı/a_-(k)a_+(k)=r_-(k)-r_+(k)=-1/a_+(k)b_+k=1/a_-(k)b_-k,

and on (ı c_,0):
    f(k)=ı/a_-(k)a_+(k)=ı (1-r_-(k)r_+(k)).

  *  If c_+>0, then |a(0)|=1, and |a(k)|≤ 1 for k∈ℝ∖{0} and |a(k)|≥1 for k∈(ı c_+,0).
  *  Nonvanishing. 
The coefficients a(k) and b( k) do not vanish on the segment (ı c_ł,ı c_);
  * (Φ^-_1(x;k))_-/a_-(k)-
      (Φ^ł_1(x;k))_+/a_+(k)=f(k)
      Φ^_2(x;k),
         k∈(ı c_ł,ı c_),

   and  (Φ^ł_2(x;k))_-/a_-
  (k)-
      (Φ^ł_2(x;k))_+/a_+(k)=
      -f(k)Φ^_1(x;k),   k∈(-ı c_,-ı
      c_ł) .Property 6.  The analytic properties of a(k) and b(k) follows from the analytic properties of the Jost solutions derived in Lemma <ref>.  The spectral coefficients a(k) and b(k) are defined by the Jost solutions by formulas (<ref>), and in turn, Jost solutions have representation (<ref>). Since E^±(x;k) have fourth-root type singularity at the points k=ı c_± (i.e. they are bounded after multiplication by (k-ı c_±)^1/4), the functions a(k) and b(k) have  at most fourth-root type singularities at ı c_±.
The analytic extension of a(k), b(k) and r(k) is discussed in the Appendix <ref>, Corollary <ref>.
Property 7.  It can be obtained by writing   the definition (<ref>)  of   the functions a(k)  and  b(k) and then using   property 1 and  the 
 large k behaviour of  the  Jost solutions in   property 4 of Lemma <ref>. 
Property 8. It  follows from the corresponding symmetries of Jost solutions (property 3 of Lemma <ref>).
Property 9. We use property 5 of Lemma <ref>, and this allows to interrelate the limiting values  T_±(k) using the definition 
Φ_±^-(x;k) = Φ^+_±(x;k)T_±(k).

For k∈(ı c_-, -ı c_-), we have Φ^-_-(x;k)=Φ^-_+(x;k)[ 0 ı; ı 0 ] and 
for k∈(ı c_+, -ı c_+),   we have   Φ^+_-(x;k)=Φ^+_+(x;k)[ 0 ı; ı 0 ]. 
Hence 
 we obtain 
    T_-(k)=[  0 -ı; -ı  0 ]T_+(k)[ 0 ı; ı 0 ],   k∈(ı c_+,-ı c_+)

and 

    T_-(k)=T_+(k)[ 0 ı; ı 0 ],   k∈(ı c_-, -ı c_+)∪ (-ı c_+,-ı c_-).

Considering the matrix entries of the above relations  completes the proof of property 9.
 Property 10. It follows  in a straightforward way from property 9.
 Property 11. Note that the absolute values of a(k) and b(k) do not have jump across the interval (ı c_-,0), i.e. |a_+(k)|=|a_-(k)|, |b_+(k)|=|b_-(k)|.
The relation (<ref>) implies |a(k)|^2+|b(k)|^2=1   for k∈ while from property 9 we have that |a(k)|^2= |b(k)|^2+1 for k∈ (ı c_+,-ı c_+).
Combining the two relations together,  we obtain that for c_->c_+> 0,  |a(k)|<1 for k∈\{0}  while |a(k)|>1 for k∈ (ı c_+,0).
For c_->c_+>0, the functions a(k) and b(k) do not have root-type singularities at the origin, and thus by continuity we obtain |a(0)|=1.
Property 12. By property 8, a(k)a( k)+b(k)b( k)=1. Taking here k on the positive side  of the  oriented  segment (ı c_-, ı c_+), we obtain a_+(k)a_+( k)+b_+(k)b_+( k)=1. By property 9, on the interval (ı c_-,ı c_+), we can express the limiting values of the spectral coefficient b in terms of the limiting values of the spectral coefficient a; thus -ı a_+(k)b_-(k) + ı a_-(k) b_+(k) = 1. By property 8, we have that the limiting values of a,b from different sides of the interval (ı c_-,ı c_+) are complex conjugates of each other, i.e. a_+(k)=a_-(k),b_+(k)=b_-(k), and hence -ı a_+(k)b_+(k) + ıa_+(k)b_+(k)=1 and (a_+(k)b_+(k)) = 1/2. Hence, neither  a(k) nor b(k) can vanish on the interval (ı c_-,ı c_+).
 Property 13. We first express Φ^+(x;k) as  Φ^+(x;k) = Φ^-(x;k)T(k)^-1,  by  formula (<ref>), and then express Φ^+_2(x;k) from the latter,

    Φ^+_2(x;k) = b( k)Φ^-_1(x;k) + a(k)Φ^-_2(x;k).

The expression on the left-hand-side does not have a jump on the contour (ı c_-, ı c_+), but every term on the right-hand-side has. Let us take the limit of the r.h.s. from the positive side of the contour (ı c_-, ı c_+), and then substitute b_+( k) = ı a_-(k) using property 11, and (Φ_2^-(x,t;k))_+=-ı (Φ_1^-(x,t;k))_- from property 5 of Lemma <ref>. This concludes the proof of the first relation. The second   relation is obtained in a similarl way.


















Let us denote the first and second columns of the Jost solution Φ^- and Φ^+ in (<ref>) as

    Φ^-_1(x;k)=[ φ^ł(x;k); ψ^ł(x;k) ],  Φ^+_2(x;k)=[ φ^(x;k); ψ^(x;k) ] ,

 so that 

    Φ^-(x;k)=[   φ^ł(x;k) -ψ^ł(x; k);   ψ^ł(x;k)  φ^ł(x; k) ],
       Φ^+(x;k)=[  ψ^(x; k)   φ^(x;k); -φ^(x; k)   ψ^(x;k) ] .

Denote the zeros of a(k) in the  quarter-plane {k∈ℂ:  k> 0
 k≥ 0} by κ_1, …,κ_N. We have κ_j∉(ı c_-,0] for j=1,…,N.  If the zeros of a(k) are simple then the residues of a^-1(k) are given by

    Res_k=κ_ja^-1(k) = ıν_j/μ_j,   j=1,…,N

where 
    (2ν_j)^-1:=∫_-∞^+∞φ^+(x;κ_j)ψ^+(x;κ_j)x̣
 and

    φ^ł(x;κ_j)=μ_jφ^(x;κ_j),  ψ^ł(x;κ_j)=μ_jψ^(x;κ_j),

with ψ^± and φ^± as in (<ref>).























Combining (<ref>) and (<ref>) we can write 
 the coefficient a(k) as

    a(k) = [ φ^-(x;k) φ^+(x;k); ψ^-(x;k) ψ^+(x;k) ],

 Let a(κ_j)=0 and ȧ(κ_j)≠0,  where ȧ(k):=ddka(k). Then, by (<ref>)
    φ^ł = μφ^,
    ψ^ł = μψ^,  μ≠0 .


If [ φ^-; ψ^- ] is a solution of the Lax equation (<ref>), then [ -ψ^-(k);  φ^-(k) ] is also a solution of the equation (<ref>).
Due to above mentioned symmetry, all the zeros κ_1,…κ_N  of a(k)  either belong to the imaginary axis or go in pairs, k and -k.
 According to property 11 and 12 of Lemma <ref>,  the zeros of a(k) are outside the segment (ı c_-, 0].


In order to prove the relation (<ref>), we differentiate in k the relation (<ref>), thus obtaining 

    ȧ(κ_j)=[   φ̇^- (κ_j) 1/μφ^- (κ_j);    ψ̇^-(κ_j)  1/μψ^-(κ_j) ]+[ μφ^+(κ_j) φ̇^+(κ_j); μψ^+(κ_j) ψ̇^+(κ_j) ].

In order to evaluate the above two determinants we  define the quantity

    W^±:=[  φ^± φ̇^±;  ψ^± ψ̇^± ],

where the dot means differentiation with respect to k.

From the Lax equation (<ref>) we have that

    φ^±_x+ı kφ^± = qψ^±,
    
    ψ^±_x-ı kψ^±=-qφ^±,
        φ̇^±_x+ı kφ̇^± + ıφ^± = qψ̇^±,
    
    ψ̇^±_x-ı kψ̇^±-ıψ^±=-qφ̇^± .


Hence, differentiating W^± in x, we obtain 

    W^±_x=2ıφ^±ψ^±,  W^±(B;κ_j)-W^±(A;κ_j)=2ı∫_A^Bφ^±(x;κ_j)ψ^±(x;κ_j) x̣

for any A<B. Choosing A=-∞, B=0 for W^- and A=0,B=+∞ for W^+, from the above relations  and (<ref>) we obtain 

    ȧ(κ_j)    =
    -1/μW^-(0;κ_j) + μ W^+(0;κ_j)
       = -1/μ∫_-∞^02ıφ^-(x;κ_j)ψ^-(x;κ_j)x̣ -μ∫_0^+∞2ıφ^+(x;κ_j)ψ^+(x;κ_j)x̣
       = -2ıμ∫_-∞^+∞φ^+(x;κ_j)ψ^+(x;κ_j)x̣ =-2ı/μ∫_-∞^+∞φ^-(x;κ_j)ψ^-(x;κ_j)x̣,

which concludes the proof of Lemma <ref>.

Note that the x-equation (<ref>) of the Lax pair is not self-adjoint, and hence  the  zeros of a(k) are not necessarily simple, and may lie everywhere in the  region  {k:  k≥ 0} as long as the symmetry relation a(- k)=a(k) is satisfied.

However, considerations similar to that of <cit.> allow to conclude that there is a dense set in the space of the step-like potentials (<ref>)  such that for any initial data  q̃(x)
satisfying (<ref>) there is an initial data q(x) close to q̃(x) 
in the topology induced by (<ref>), for which the 
 generic Assumptions  <ref> are satisfied.
 
Summarizing we arrive to the following set of scattering data for the initial data  q(x,0) satisfying the Assumptions <ref>  and  <ref>:

    𝒮={r(k), {κ_j,ıν_j}_j=1^N}

with r(k) meromorphic   in a δ-neighbor­hood of Σ=ℝ∪[ı c_ł,-ı c_ł], where δ = √(σ^2+c_+^2)-c_- and  σ>√(c_-^2-c_+^2)>0
and where  {κ_j}_j=1^N  are  simple zeros of a(k) with  κ_j∈^+\{∪(ı c_-,0]}  and κ_j≥ 0.
If κ_j corresponds to a soliton, then κ̅_j is also a point of the discrete spectrum, while if κ_j corresponds to a breather, then -κ_j, κ_j and -κ_j belong to the discrete spectrum.







 



 §.§ Riemann-Hilbert problem in the generic case


In this section we introduce the RH  problem <ref>, which allows to reduce the (nonlinear) initial value problem (<ref>), (<ref>) into a (linear) matrix conjugation problem.




In order to set up  a  RH problem for  t≥0  we proceed as follows.
We first assume that the solution q(x,t) of the initial value problem (<ref>), (<ref>) exists, and, moreover, that 
 the Jost solutions Φ^±(x,t;k)  with the defining property 
 
    Φ^±(x,t;k)=E^±(x,t;k)(1+o(1))  ,   k∈ℝ∪[ı c_±, -ı c_±],

  exist  for t≥ 0 (here, the matrix E^±(x,t;k) has been defined in (<ref>)).  
Our assumption will be dropped off later, after deriving the RH problem <ref>.
We will show the solvability of the RH  problem  <ref>  and thus we  justify a posteriori the assumption of existence of q(x,t) and  the  Jost solutions Φ^±(x,t;k).

 To start, we observe   that since Φ^±(x,t;k) are solutions to the  first order equation (<ref>),
  these solutions are related by the  linear transformation
  
    Φ^ł(x,t;k)=Φ^(x,t;k)T(t;k),  
    k∈ℝ∪(ı c_,-ı c_).

Using the evolution equation <ref>, it follows that

    ddtT(t;k)=0,

namely ddta(k,t)=0 and ddtb(k,t)=0.  The fact that the  scattering data are constant in time  is due to our choice of normalization 
of  the  Jost  solutions in (<ref>).


The scattering relations (<ref>) between the matrix-valued
functions Φ^ł(x,0;k) and
Φ^(x,0;k),  and the jump conditions  of Lemma <ref> and Lemma <ref> can
be written as a matrix Riemann – Hilbert problem (RHP). Namely, let us notice that the matrix-valued function

    M(x,0;k)=
    (
    Φ^ł_1 (x,0;k)/a(k) e^ı
    kx,Φ^_2(x,0;k)e^-ı kx),
      k∈ℂ_+ \ [ c_ł,0],  
    
    (Φ^_1(x,0;k)e^ı kx,Φ^ł_2(x,0;k)/a(k)
    e^-ı kx), k∈ℂ_-\[0, -
    c_ł],
 
satisfies the jump conditions 

    M_-(x,0;k)=M_+(x,0;k)J(x,0;k),   k∈Σ,

where  the matrix J(x,0;k) and the oriented contour  Σ=∪[ı c_-,-ı c_-]
are specified in  Figure <ref>. 
 Here M_±(x,0;k) are the limiting values of the matrix M(x,0;k) as k approaches the contour from the positive/negative direction (the positive direction is on the left, the negative is on the right as one traverses the contour in the direction of orientation). Using Lemmas <ref> and <ref> the matrix J(x,0;k) can be obtain in a straightforward way from the definition (<ref>)<cit.>.
 Further, let us assume that κ_j with κ_j≥ 0 and κ_j> is a zero of a(k). Then taking the residue at k=κ_j of the matrix M(x,0;k)  and using  Lemma <ref>
 we obtain
 
    Res_κ_jM(x,0;k)=(
    ıν_j/μ_jΦ^ł_1 (x;κ_j) e^ı kx,0)=lim_k→κ_jM(x,0;k)[                 0                 0; ıν_j^2ıθ(x,0;κ_j)                 0 ],

and similarly for κ̅_j, -κ_j and -κ̅_j.



The jump properties of the matrix M(x,0;k) prompts us to consider the following  matrix valued function

    M(x,t;k)=
    (
    Φ^ł_1 (x,t;k)/a(k) e^ıθ(x,t;k),Φ^_2(x,t;k)e^-ıθ(x,t;k)),
      k∈ℂ_+ \ [ c_ł,0],  
    
    (Φ^_1(x,t;k)e^ıθ(x,t;k),Φ^ł_2(x;k)/a(k)
    e^-ıθ(x,t;k)), k∈ℂ_-\[0, -
    c_ł],
 
where 
    θ(x,t;k)=xk+4k^3t.
 
Using  the properties of the Jost solutions described in section <ref>, we can check that the function M(x,t;k), (assuming it  exists),  satisfies the jump, analyticity, and normalization conditions described below in the RH problem <ref>.










To find a 2× 2 matrix-valued function M(x,t;k) with the following properties:

  * M(x,t;k) is meromorphic  for  k∈ℂ∖Σ,  where  Σ=∪[ı c_-,-ı c_-]   (see Figure <ref>) and  it has at most fourth root singularities at the points ±ı c_±. 
  
  *  The boundary values M_±(x,t;k)  on the oriented contour Σ satisfy the jump condition
   
    M_-(x,t;k)=M_+(x,t;k)J(x,t;k),    k∈Σ

    and  

    J(x,t;k)={[ [                 1 -r(k)^-2iθ(x,t;k);  -r(k)^2ıθ(x,t;k)        1+|r(k)|^2 ],                                                                      k∈ℝ∖{0},                                                                              ;         [               1               0; f(k)^2ıθ(x,t;k)               1 ],                                                               k∈(ı c_ł,ı c_),;                                                                              ;     [         ı r_-(k)    ı^-2ıθ(x,t;k); f(k) ^2ıθ(x,t;k)        -ı r_+(k) ],                                                                   k∈(ı c_,0),;                                                                              ; [           ır_+(k) -f(k)^-2ıθ(x,t;k);      ı^2ıθ(x,t;k)          -ır_-(k) ],                                                                  k∈(0,-ı c_),                                                                              ; [                 1 -f(k)^-2ıθ(x,t;k);                 0                 1 ],                                                             k∈(-ı c_,-ı c_ł),                                                                               ].

where r(k) is the reflection coefficient  of the  spectral problem of the  MKdV  equation, and
 
    f(k)=r_-(k)-r_+(k)=ı/a_-(k)a_+(k),   k∈(ı c_ł,ı c_)∪ (-ı c_,-ı c_ł),
a^-1(k) is the transmission coefficient and 
 
    θ(x,t;k)=kx+4k^3t .

  *  Simple poles: residue condition at κ_j  and -κ_j for j=1,...,N with κ_j≥ 0 and κ_j>0:
    Res_κ_jM(x,t;k)=lim_k→κ_jM(x,t;k)[                 0                 0; ıν_j^2ıθ(x,t;κ_j)                 0 ],

    Res_-κ_jM(x,t;k)=lim_k→ -κ_jM(x,t;k)[                0                0; ıν_j^2ıθx,t;-κ_j                0 ];

 residue conditions  in the lower half-plane:

    Res_κ_jM(x,t;k)=lim_k→κ_jM(x,t;k)[                   0 ıν_j^-2ı tθx,t; κ_j;                   0                   0 ],

    Res_-κ_jM(x,t;k)=lim_k→-κ_jM(x,t;k)[                   0 ıν_j^-2ıθ(x,t;-κ_j);                   0                   0 ];

  *  asymptotics: M(x,t;k)→ I as k→∞.

This is the point at which we drop off the assumption of existence of q(x,t) and Φ^±(x,t;k),
 we study the RH problem <ref> and its solvability. 
Note that  the  jump matrix J(x,t;k) is such that the  matrix M(x,t;k) in  (<ref>) has a trivial monodromy   at the origin.
Further we observe that the jump matrix J(x,t;k) satisfies the  Schwartz symmetry J^-1(k)=J^T(k) for k∈Σ∖ℝ.[We observe that the usual definition of Schwartz symmetry
refers to a contour Σ such that Σ=Σ and the corresponding jump matrices satisfy the relation J^T(k)=J(k).
In our case, we orient the contour off the real axis in such a way that it is symmetric up to orientation, this  implies that J^T(k)=J^-1(k) for k off the real axis.
]

The solution M(x,t;k)=M(k) of the RH problem <ref> automatically satisfies the following symmetries:

    M(k) = M(- k) = [  0 -1;  1  0 ]
    M(-k)
    [  0  1; -1  0 ]=[  0 -1;  1  0 ]M( k)[  0  1; -1  0 ] .
The existence and smoothness of a solution to  RH problem (<ref>) for the case c_+=0 was established in <cit.>. The  case c_+>0 can be treated similarly; below we give the details.

The existence of solution of the RH problem <ref> is based on the vanishing lemma for Schwartz symmetric RH problems, <cit.>. In our case, because of the choice of orientation of the parts of the contour off the real axis, it reads as 
J^-1(k)=J^T(k) for k∈Σ∖ℝ, and J(k)+J^T( k) is positive definite for k∈ℝ.



Further, from   our assumption on the initial data it follows the analyticity of the reflection coefficient  in a small neighbourhood of the contour Σ, (see Lemma <ref>). Therefore  we can deform the contour Σ into  a   new contour such that
the corresponding singular integral equation, which is equivalent to the RH problem <ref>, 
admits differentiation with respect to x and t.
Then, in the spirit of the well-known result of Zakharov – Shabat <cit.>, one can prove that 
the solution of the initial value problem (<ref>), (<ref>)  exists and  can be reconstructed by the following formula (see <cit.>, chapter 2 for details):

    q(x,t)=lim_k→∞2ı k M(x,t;k)_21=lim_k→∞2ı k M(x,t;k)_12.



Note that in the framework of the RH problem with unimodular jump matrix and with locally L_2 integrable boundary values of solutions,  one fixes the value of  the  solution at a given point (∞ in our case), and specifies all the possible poles of the solution. This fixes the solution uniquely. Indeed, first of all, the determinant of a solution does not have jump over the contour, tends to the identity at infinity, and has at most removable singularities at the points ±ı c_±; hence it equals 1 identically. Second,  assuming that there are two solutions of the RH problem, their ratio would have trivial jumps and will tend to identity at infinity, and hence would be equal to identity (see, for instance <cit.>).





In the case when the contour has points of self-intersections, the condition of continuity of the jump matrix is replaced at the points of self-intersection by the condition that the product of jump matrices equals identity matrix at the points of self-intersection. This guarantees that the solution takes its limiting values continuously from within each of the domains separated by the contour. See <cit.> and <cit.> for more detail.



















In order to make the asymptotic analysis  of the above RH problem as t→∞ 
 it is more convenient to transform the residue conditions at the poles to a jump conditions as in <cit.>.


Let κ_j be a pole 
of a(k).  Let us encircle this pole with a  circle C_j  of radius ε>0.
There are several options:


  * κ>0, κ>0. In this case C_j is a  circle  of radius ε and center   κ_j  and oriented anticlockwise  with respect to the center. Let us also define the three other
circles C_j,-C_j,-C_j  with center  the points κ_j,-κ_j, -κ_j,  and radius ε and oriented anticlockwise.





  * κ_j=0,  κ_j>0. 
  In this case we denote C_j a semicircle in the  plane k≥0 around the point κ_j,
 while -C_j is another semicircle in
k≤0 around the point κ_j. Semicircles C_j and -C_j are the corresponding semicircles in the lower half-plane, with the above agreement on the orientation.




















We replace the residue condition with a jump condition having only upper triangular matrices. For the purpose we    redefine the matrix M(k) as  

    M(k) M(k)[                          1                          0; -ıν_j^2ıθ(x,t;κ_j)/(k-κ_j)                          1 ],    k  is inside  C_j,

    M(k) M(k)[                            1                            0; -ı ν_j^2ıθ(x,t;-κ_j)/(k+κ_j)                            1 ],    k  is inside  -C_j,

    M(k) M(k)[                             1 -ı ν_j ^-2ıθ(x,t;κ_j)/(k-κ_j);                             0                             1 ],    k  is inside C_j,

    M(k) M(k)[                             1 -ıν_j ^-2ıθ(x,t;-κ_j)/(k+κ_j);                             0                             1 ],    k  is inside  -C_j.

Then the  jump matrix J_M(k) for the RH  problem   for M becomes

    J_M(k)=[                         1                         0; ıν_j^2ıθ(x,t;κ_j)/(k-κ_j)                         1 ],    k ∈ C_j,  
    [                           1                           0; ı ν_j^2ıθ(x,t;-κ_j)/(k+κ_j)                           1 ],    k ∈-C_j,
    [                            1 ı ν_j ^-2ıθ(x,t;κ_j)/(k-κ_j);                            0                            1 ],    k ∈C_j,
    [                            1 ıν_j ^-2ıθ(x,t;-κ_j)/(k+κ_j);                            0                            1 ],    k ∈ -C_j,
    
               J(k),

where J(k) has been defined in (<ref>).




 §.§ Solvability of RH problem and existence of solution for the MKdV
Denote by C^∞(ℝ_x×ℝ_t^+) 
the set of functions, which are infinitely many times differentiable at any point (x,t)∈ℝ_x×ℝ_t^+,
where ℝ_t^+={t: t∈(0,+∞)}.

Let the  initial data q_0(x)(<ref>) satisfiy  Assumption <ref>.




Then the initial value problem (<ref>), (<ref>) has a unique classical solution q(x,t), which  belongs to the class
C^∞(ℝ_x×ℝ^+_t). The proof is similar to the one in <cit.>, <cit.>, except that we need to make an extra step in order to treat the discontinuity of the jump matrices at the points ±ı c_±. For the convenience of the reader, we will sketch the main steps.
 
Step 1. Transforming the RH problem to another one without points of discontinuities.

Let C_0 be a contour which encloses the segment [ı c_-,-ı c_-], do not pass through any poles of M, and lies in the domain of analyticity of the reflection coefficient r(k) (see the left figure  in
 Figure <ref>). Let C_0∩ℝ={-k_1, k_1}.
Let Ω_+, Ω_- be the domains inside C_0 which lie inside ℂ_±={k:± k>0}, respectively.
We transform the RH problem <ref> for the  function M(x,t;k) to an equivalent RH problem for a matrix function M^(1)(x,t;k), where the functions M and M^(1) are related as follows: M^(1)(x,t;k)=M(x,t;k)[                1                0; -r(k)^2ıθ(x,t;k)                1 ]
for k∈Ω_+,M^(1)(x,t;k)=M(x,t;k)[                 1 r( k)^-2ıθ(x,t;k);                 0                 1 ] for k∈Ω_-,
and M^(1)(x,t;k)=M(x,t;k) elsewhere.
Property <ref> of Lemma <ref> allows to verify that the jump for M^(1) on the intervals (ı c_-,ı c_+) and (-ı c_+, -ı c_-) becomes identity, and on the interval (ı c_+,-ı c_+) it becomes [             0 ı^-2ıθ(x,t;k);  ı^2ıθ(x,t;k)             0 ].
The next transformation removes also the jump on [ı c_+,-ı c_+]; this is done by passing to a function M^(2)(x,t;k), which is related to M^(1) in the following way:
M^(2)(x,t;k) = M^(1)(x,t;k)M_C(x,t;k)^-1 for k∈Ω_+∪Ω_-,
and 
M^(2)(x,t;k)=M^(1)(x,t;k) elsewhere.
Here 
    M_C(x,t;k) = 1/2[               γ+γ^-1 (γ-γ^-1)^-2ıθ(x,t;k);  (γ-γ^-1)^2ıθ(x,t;k)               γ+γ^-1 ]
 with γ=γ(k)=√(k-ı c_+/k+ı c_+).
The matrix M^(2)(x,t;k) is a solution of a RH problem with the jump matrix J^(2)(x,t;k),M^(2)_-(x,t;k)=M^(2)_+(x,t;k)J^(2)(x,t;k),  for 
k∈Σ_2 = ℝ∪ C_0\(-k_1,k_1).
 Note that since the singularities of M and M_C at the points ±ı c_± are bounded by (k∓ı c_±)^-1/4, and since M≡ M_C ≡ 1, the  matrix  M^(2)(k) has singularities at these points which are bounded by 
(k∓ı c_±)^-1/2. Since  M^(2)(k)  does not have a jump on  [±ı c_-,±ı c_+] it follows  that the points ±ı c_±  are removable singularities.




Step 2. Solvability of RH problem for M^(2).
The solvability of the RH problem <ref> is thus reduced to solvability of the RH problem for the function M^(2). 
We introduce the operator 

    𝒦[f](k)= 1/2πı∫_Σ_totf(s)(I-J^(2)(x,t;s))ṣ/(s-k)_+,   f∈ L_2(Σ_tot;ℂ^2× 2)

 where the symbol (s-k)_+  stands for the  limiting value of the integral on the positive side of the oriented contour 
 
    Σ_tot=ℝ∪ C_0 ∪_j C_j∪_jC_j∪_j- C_j∪_j-C_j\(-k_1,k_1).

 The operator 𝒦 is a  bounded linear   operator from L_2(Σ_tot;ℂ^2× 2) to itself.
 

 It is standard <cit.>, that the RH problem 
 for M^(2) is equivalent to the following singular integral equation for the  matrix  function μ=M^(2)_+-I:
    μ(x,t;k)=𝒦[μ](x,t;k)+𝒦[I](x,t;k).
 Further, the jump matrix J^(2)(x,t;k) off the real axis is Schwartz reflection invariant<cit.>(note the inverse power of the jump matrix in the second property below, which is due to  the fact that the orientation of the contour off the real axis changes to the opposite one under the map k↦k, see Figure <ref>):

  *  the contour Σ_tot is symmetric with respect to the real axis ℝ(up to the orientation),
  
  * J^(2)(x,t;k)^-1=J^(2)(x,t;k)^T for k∈Σ_tot∖ℝ
  * J^(2)(x,t;k) has a positive definite real part for k∈ℝ.
Then Theorem 9.3 from <cit.>(p.984) guarantees that the operator 
Id-𝒦, where Id is the identity operator,   is invertible as an operator acting from  L_2(Σ_tot;ℂ^2× 2) to itself.  
  Invertibility is guaranteed from the fact that Id-𝒦  is a Fredholm integral operator with zero index and the kernel
of Id-𝒦 is  the zero 2× 2 matrix.  In particular this last point is obtained by applying the vanishing lemma. Indeed suppose that exists μ∈ L_2(Σ_tot;ℂ^2× 2) such that
 (Id-𝒦)μ=0. Then the quantity
 
    M_0(x,t;k)=1/2πı∫_Σ_totμ(s)(I-J^(2)(x,t;s))ṣ/s-k

solves the  following  RH problem:
  (1)M_0(x,t;k)  is analytic in \Σ_tot,  

  (2)M_0-(k)=M_0_+(k)J^(2)(k) for k∈Σ_tot,

  (3)M_0(k)=𝒪(k^-1) as k→∞.  

We define  H(k)=M_0(k)M_0(k)^T, then clearly H(k)^T=H(k), H(k)=𝒪(k^-2)  as k→∞  and H(k) is analytic in ^+\{C_0∪_jC_j∪_j-C_j}.
 Following <cit.>, we integrate the function H(k) over the boundary of every closed component in ℂ_+∖Σ_tot (those components are separated by contours C_0 and C_j; each component is integrated in  the  positive direction), and then add them to each other. Integrals over each part of the boundary except for the real axis is taken twice, and 
we thus obtain that by analyticity 

    ∫_-∞^+∞H_+(k)dk
    -
    ∫_C_0∩ℂ^+(H_-(k)-H_+(k))dk
    -
    ∑_j(∫_C_j
    +
    ∫_-C_j)(H_-(k)-H_+(k))dk
    
       =∫_-∞^∞ M_0+(k)J^(2)(k)^TM_0+(k)^Tdk
    -
    ∫_C_0∩ℂ^+M_0+(k)(J^(2)(k)J^(2)(k)^T-I)M_0+(k)^Tdk
           -∑_j(∫_C_j+∫_-C_j)M_0+(k)J^(2)(k)J^(2)(k)^T-IM_0+(k)^Tdk=0 .


Here we defined J^(2)=I for k∈ (-k_1, k_1), the part of the real axis not in Σ_tot.
To obtain the second identity we use the jumps J^(2)(x,t;k) of the function M_0.
The symmetry properties of the jump matrix J(x,t;k) imply that J^(2)(k) = J^(2)( k)^-1 for k∈Σ_tot\ℝ, and hence the integrals over the  contour C_0∩ℂ^+ and the circles C_j and -C_j do not give any contribution, so that one obtains
 
    ∫_-∞^∞ M_0+(k)(J^(2)(k)+J^(2)(k)^T)M_0+(k)^Tdk=0 .

Since J^(2)(k)+J^(2)(k)^T is positive definite, it follows that M_0+(k) is identically zero.  This shows that the operator  Id-𝒦  has a trivial kernel.   


Therefore, the singular integral equation (<ref>) has a unique solution M^(2)_+(x,t;k)-I∈ L_2(Σ_tot;ℂ^2× 2) for any fixed
x,t∈ℝ, and the matrix M^(2)(x,t;k)

can be obtained by the formula 
    M^(2)(x,t;k)=I+1/2πı∫_Σ_totM^(2)_+(x,t;s)(I-J(x,t;s))ṣ/s-k.

Finally, inverting the transformations that led from the matrix M(x,t;k) to the matrix M^(2)(x,t;k) in Step 1, we obtain the 
solution of the  RH problem <ref>. 
Step 3. Differentiability of M(x,t;k) with respect to x,t.
First of all we notice that it is impossible to differentiate the equation (<ref>) with respect to x,t, since the 
function r(k), as well as the matrix I-J(x,t;k), vanishes as k^-1 as k→∞ along the real axis. 
To avoid a weak rate of decreasing of the matrix I-J(x,t;k) for large real k, we use an equivalent RH problem where the jump matrix J(x,t;k) for large k becomes exponentially close to I.















































 

Let take a positive K>0. Then for k>K  we use the following factorization 
of the jump matrix:

    J(x,t;k)=[                 1                 0; -r(k)^2ı t θ(k,ξ)                 1 ][                  1 -r(k)^-2ı t θ(k,ξ);                  0                  1 ],
 which allows to transform the RH problem <ref>  for the matrix M(x,t,;k)  into the  one for the  matrix M(x,t;k), 
where the parts of the contour (-∞, -K),(K,+∞) are splitted into two lines in the upper and lower complex planes (see the right figure of Figure <ref>).
As a result, the jump J(x,t;k) of the transformed RH problem is exponentially close to I for large k on the transformed contour 
Σ_tot.
The corresponding singular integral equation is as follows for the matrix μ=M_+-I:

    μ(x,t;k)=𝒦[μ](x,t;k)+𝒦[I](x,t;k),

where 𝒦 is as 𝒦 in (<ref>) with  J  replaced by J,
Following the  reasoning  as for (<ref>), the above integral equation has a unique solution in  L_2(Σ_tot;ℂ^2× 2).
Equation 
(<ref>) has the advantage, that we can differentiate it with respect to x,t as many times as we wish.
Indeed, while the function I-J(x,t;k) vanishes as 1/k on ℝ, the  function I-J(x,t;k) decays exponentially  fast on the infinite parts of the contour 
Σ. The singular integral equation, obtained by differentiating with respect to t(<ref>), has the same form as 
(<ref>) (only the right-hand-side of this equation changes). It provides unique solvability of the partial derivatives of 
μ(x,t;k) 	with respect to x,t. Hence, the same is true for M(x,t;k).Step 4. Zakharov-Shabat scheme.
It is a standard Lax-pair argument (see, for instance, <cit.>) to show, that the function 

    Φ(x,t;k)= M(x,t;k) ^(ı k x+4ı k^3 t)σ_3

satisfies the Ablowitz-Kaup-Newell-Segur system of equations  (see e.g. <cit.>)

    Φ_x(x,t;k) +ı kσ_3Φ(x,t;k)=Q(x,t)Φ(x,t;k),
    
    Φ_t(x,t;k) +4ı k^3σ_3Φ(x,t;k)=Q(x,t;k)Φ(x,t;k),
        
    x∈ℝ, t>0,

where 
    Q(x,t)=[       0  q(x,t); -q(x,t)       0 ],

    Q(x,t;k)=4k^2Q(x,t)-2ı kQ^2(x,t)+Q_x(x,t)σ_3+2Q^3(x,t)-Q_xx(x,t)

with the function q(x,t) given by 

    q(x,t)=2ılim_k→∞k[M(x,t;k)]_21=-1/π∫_Σ(I+μ(x,t;k))I-J(x,t;k)ḳ.


§ MODEL PROBLEMS

The asymptotic analysis of the Cauchy problem as t→∞ consists of several steps:

  *  the first step is to change the original phase function, which is present in the exponents of  the RH problem <ref>, with an appropriate  function g(k,ξ) that will be determined later; 

  *  the second step consists in performing a chain of` “exact" transformations of the RH problem;

  *  the third step consists in approximating  the new RH problem  to some  model problem;

  *  the fourth step consists in solving the model problems.

Before starting our asymptotic  analysis, we  introduce the RH  model problems that will   be obtained  from such analysis.
Namely the  RH problem for the soliton and breather solution on a constant background and the RH problem for the travelling wave solution (<ref>). In particular for the travelling wave solution, we show
that  it can be obtained from a model problem solvable in terms of elliptic theta functions but also from a model problem that is solvable via hyperelliptic theta functions
that are defined on a genus 2 hyperelliptic Riemann surface with symmetries. The first case arises when the step-like initial data is such that c_=0, while the second case occurs when c_>0.
 Then  we show that the genus 2  solutions can nevertheless be written in a genus 1 form.



 §.§ One-soliton solution on a constant background c>0

Here we derive a one-soliton solution on a constant background. We use the notation [ı c, -ı c] to denote the interval oriented dowward.



 Find a 2× 2 matrix M(k)=M(x,t;k)  meromorphic for k∈ℂ∖[ı c, -ı c] with simple poles at k=±ıκ_0,   κ_0>c>0,  with the following properties:

  *  the boundary values M_±(k) for k∈ (ı c, -ı c) satisfy the  jump relations
 
    M_-(k)=M_+(k)[ 0 ı; ı 0 ],   k∈(ı c, -ı c);

  *  pole conditions: 
 
    Res_ıκ_0M(k)=lim_k→ıκ_0M(k)[                 0                 0; ıν^2ı  g_c(x,t;k)                 0 ],
       Res_-ıκ_0M(k)=lim_k→ -ıκ_0M[                  0 ıν^-2ı  g_c(x,t;k);                  0                  0 ],

where  ν is a non zero real constant and  

    g_c(k; x,t)=(2(2k^2-c^2)t+x)√(k^2+c^2);

  *  asymptotics: 

    M(k)= I+O(1/k),
 The solution of the MKdV equation    is obtained from   the matrix M(k;x,t) by the relation 

    q(x,t)=2ılim_k→∞k M_21(x,t;k)=2ılim_k→∞k M_12(x,t;k).

The solution of   the MKdV q_t+6q^2q_x+q_xxx=0 obtained from the solution of the  RH problem  <ref>  is equal to  a soliton on a constant background c, namely

    q_soliton(x,t; c,κ_0,ν)=c-2 sgn(ν)(κ_0^2-c^2)/κ_0cosh[2√(κ_0^2-c^2)(x-(2c^2+4κ_0^2)t)+x_0]-sgn(ν)c,

where

    x_0=ln2(κ_0^2-c^2)/|ν|κ_0 .


We first obtain the solution of  the RH problem  described by equations (<ref>) and (<ref>) in the form

    M_reg(k)=[ ψ_2(k) ψ_1(k); ψ_1(k) ψ_2(k) ],

where 
 
    ψ_2=12(√(k-ı c/k+ı c)+√(k+ı c/k-ı c)),  ψ_1=12(√(k-ı c/k+ı c)-√(k+ı c/k-ı c)).


Since M(k) M_reg(k)^-1 does not have jumps on  but only poles, 
 the solution  M(k)  can be found in the form

    M(k;x,t)=
    [ 1+ıα(x,t)/k-ıκ_0+ıγ(x,t)/k+ıκ_0  ıβ(x,t)/k+ıκ_0+
    ıδ(x,t)/k-ıκ_0;   ıβ(x,t)/k-ıκ_0+ıδ(x,t)/k+ıκ_0 1-ıα(x,t)/k+ıκ_0-ıγ(x,t)/k-ıκ_0 ]
    M_reg(k),

where α, β, γ, δ are real parameters to be determined.
The solution of the MKdV equation    is obtained from   the matrix M(k;x,t) through the formula

    q(x,t)=2ılim_k→∞k M_21(x,t;k)=2ılim_k→∞k M_12(x,t;k)=c-2β-2δ.

We observe that   due the symmetry of the problem, it is enough to consider the residue condition only at one of the poles k=±ıκ_0.   For example the condition <ref> at   k=ıκ_0  gives 

    αψ_1(ıκ_0)+δψ_2(ıκ_0)=0,
       βψ_1(ıκ_0)-γψ_2(ıκ_0)=0,
       1ν(αψ_2(ıκ_0)+δψ_1(ıκ_0))^-2ı g_c(ıκ_0)=ψ_1(ıκ_0)+γ2κ_0ψ_1(ıκ_0)+
       +
    β2κ_0ψ_2(ıκ_0)+ıαψ_1'(ıκ_0)+ıδψ_2'(ıκ_0),
       1ν(βψ_2(ıκ_0)-γψ_1(ıκ_0))e^-2ı g_c(ıκ_0)=ψ_2(ıκ_0)+δ2κ_0ψ_1(ıκ_0)-
       -α2κ_0ψ_2(ıκ_0)+ıβψ_1'(ıκ_0)-ıγψ_2'(ıκ_0),

where ' stands for derivative with respect to k.  Solving the above system of equations we obtain 

    α=-δψ_2(ıκ_0)ψ_1(ıκ_0),  γ=βψ_1(ıκ_0)ψ_2(ıκ_0)




and 






    β(x,t)=√(κ_0^2-c^2)ν^2 i g_c{-cνκ_0 ^2ı g_c+2(κ_0^2-c^2)(κ_0+√(κ_0^2-c^2) ) }/{4(κ_0^2-c^2)^2-4(κ_0^2-c^2)ν c ^2ı g_c+κ_0^2ν^2^4ı g_c},

    δ(x,t)=√(κ_0^2-c^2)ν^2 i g_c{cνκ_0 ^2ı g_c-2(κ_0^2-c^2)(κ_0-√(κ_0^2-c^2) ) }/{4(κ_0^2-c^2)^2-4(κ_0^2-c^2)ν c ^2ı g_c+κ_0^2ν^2^4ı g_c},

where g_c=g_c(iκ_0).








Plugging the above expressions for β and δ into (<ref>) we obtain the statement of the lemma.






We observe that when c=0, the formula (<ref>) coincides with the one-soliton solution (<ref>).

Let us notice that the denominator in the  formula (<ref>) is always nonzero.






Since κ_0>c, in the case ν>0 we have antisoliton, oriented downward, and for ν<0 we have a  soliton, oriented upward.

We see that for κ_0-c→+0 and ν<0 the amplitude of the soliton tends to 0, while for ν>0 the amplitude oantisolitonntisoliton tends to a constant 2κ.Degenerate case of antisoliton.

When ν>0 and

 we let k_0→ c+0, we obtain the special case of an antisoliton,  namely a  rational solution.

In this case 








    q(x,t; c,c,ν)=c-4c/1+(2c(x-x_0)-12c^3t)^2.

Let us observe that the MKdV admits a RH problem with poles of higher order <cit.>. This case  is  non-generic and it is considered in  Appendix <ref>.
 

 §.§ Simple breathers on a constant  background

In this section we consider a breather on a constant  background.


 Find a 2× 2 matrix  M(x,t;k)  meromorphic in k∈ℂ\[-ı c,ı c] with simple poles at κ≡κ_1+iκ_2,κ≡κ_1-iκ_2,-κ≡-κ_1-iκ_2,-κ≡-κ_1+iκ_2, and such that 

  * M_-(k)=M_+(k) J(k),   k∈(ı c,-ı c), where 
    J(k)=[ 0 ı; ı 0 ],   k∈(ı c,-ı c);

  *  pole conditions: 
    Res_κM(k)=lim_k→κM(k)[                  0                  0; ıν^2ı   g_c(x,t;k)                  0 ],

    Res_-κM(k)=lim_k→ -κM(k)[                   0                   0; ı ν^2ı   g_c(x,t;k)                   0 ],

    Res_κM(k)=lim_k→κM(k)[                   0 ı ν^-2ı  g_c(x,t;k);                   0                   0 ],

    Res_-κM(k)=lim_k→ -κM(k)[                 0 ıν^-2ı g_c(x,t;k);                 0                 0 ],

where ν is a non zero complex number and 
g_c(x,t;k) as in (<ref>);
  *  asymptotics: M(k)=I+O(1/k) as k→∞.




The solution of MKdV q_t+6q^2q_x+q_xxx=0 is obtained from the solution of this RH problem by one of the following formulas:

    q_breather(x,t)=2ılim_k→∞k M_21(x,t;k)=2ılim_k→∞k M_12(x,t;k),

or

    q_breather^2(x,t)=c^2+2ı∂_x lim_k→∞ k(M-I)_11.

The solution (<ref>) of the MKdV equation  obtained from the solution of the RH problem  <ref> corresponds to a breather 
 on a constant background c   with discrete spectrum κ=κ_1+ıκ_2 and complex parameter ν  and  takes the form  
    q_breather(x,t;c,κ,ν) =c+
       2∂_xarctan[|χ|cos( 2 g_c(x,t)+θ_1-θ_2)+c|ν| χ_1^2/2|χ|^2 χ_2^- 2 g_c(x,t)|χ|^2/|ν|^2 g_c(x,t)+χ_1^2(|χ|^2-c^2)/4|χ|^2χ_2^2|ν|^ -2 g_c(x,t)+c sin( 2 g_c+θ_1-2θ_2)]
 
where  χ=χ_1+ıχ_2:==√(κ^2+c^2),  with  χ_1>0, χ_2>0 and  real and 
 
    g_c(x,t)=χ_1(4(χ_1^2-3χ_2^2-3/2c^2)t+x),
        g_c(x,t)=χ_2(4(3χ_1^2-χ_2^2-3/2c^2)t+x),

 and  phases θ_1=-ν_2|ν| and θ_2=χ_1|χ|.
 Formula (<ref>) coincides with formula     (<ref>)   for the breather provided in the introduction. 
 
Let us introduce the quantities
 
    Z=x+4t(3χ_1^2-χ_2^2-3/2c^2),   φ=2(Z-8t|χ|^2)χ_1+θ_1-θ_2.

Then  the breather (<ref>) on the constant background c can be written in the form
 
    q_breather(x,t;c,κ,ν)=c+
       +2∂_xarctan[|χ|cos(φ)+c|ν| χ_1^2/2|χ|^2 χ_2^- 2 Z χ_2√(|χ|^2-c^2)χ_1χ_2cosh ( 2 Z χ_2+θ_3) +csin(φ-θ_2)],   if   |χ|>c,
    ∂_xarctan[|χ|cos(φ)+c|ν| χ_1^2/2|χ|^2 χ_2^- 2 Z χ_2√(c^2-|χ|^2)χ_1χ_2sinh ( 2 Z χ_2+θ_3) +csin(φ-θ_2)],   if    |χ|<c,
    ∂_xarctan[cos(φ)+|ν| χ_1^2/2|χ|^2 χ_2^- 2 Z χ_2c/|ν|^2Zχ_2+sin(φ-θ_2)], if    |χ|=c,

where θ_3=log2χ_2|χ|^2/|ν|χ_1√(|χ|^2-c^2).
 
The solution  of the RH problem <ref>
 can be obtained  in the form

    M(k)=M_pol(k)M_reg(k),
 where 
M_reg(k)   has been defined in (<ref>) and M_pol(k)=M_pol(x,t;k) admit an ansatz of the form 

    M_pol(k)=I+
    [ A(x,t)/k-κ - A(x,t)/k+κ + C(x,t)/k+κ - C(x,t)/k-κ    B(x,t)/k+κ-B(x,t)/k-κ -D(x,t)/k+κ + D(x,t)/k-κ; B(x,t)/k-κ - B(x,t)/k+κ + D(x,t)/k+κ - D(x,t)/k-κ A(x,t)/k-κ - A(x,t)/k+κ + C(x,t)/k+κ - C(x,t)/k-κ ],

where A=A(x,t), B=B(x,t), C=C(x,t) and D=D(x,t) are unknown functions to be determined.







Writing down the pole conditions at the point κ,κ>0,κ>0, which is enough due to symmetries of the problem, we obtain 
 the following system of equations:

    Aψ_1+Dψ_2=0,
    
    Bψ_1-Cψ_2=0,      (pole free condition for the 2nd row at κ)
    
    Aψ_2+Dψ_1/ıν^2ı g_c=ψ_1(1-A/κ+κ +C/2κ -C/κ-κ ) + 
    ψ_2(-B/κ-κ+B/2κ-D/κ+κ)+ψ_1'A+ψ_2'D,
    
    Bψ_2-Cψ_1/ıν^2ı g_c=ψ_1(-B/κ+κ+D/2κ-D/κ-κ)+ψ_2(1+A/κ-κ-A/2κ+
    C/κ+κ)+ψ_1' B-ψ_2' C,

where   we use the compact notation ψ_j=ψ_j(κ)  and  ψ_j'=ψ_j'(κ),  where ψ_j'(κ)=ddkψ_j(k)|_k=κ 
and
 
    ψ_1(k)=12(√(k-ı c/k+ı c)-√(k+ı c/k-ı c)),  ψ_2(k)=12(√(k-ı c/k+ı c)+√(k+ı c/k-ı c)).

Let us introduce 

    ℛ(k)=ψ_1(k)/ψ_2(k)=ıc(k-√(k^2+c^2)).

With the above notation it is straighforwad to obtain the solution  (<ref>) of the MKdV equation 
 in the form
 
    q_breath(x,t)=c+4(A(x,t)ℛ(κ)-B(x,t)).

 We need to determine the quantities A(x,t) and B(x,t) in (<ref>). These constants are obtained by solving the system of 
equations (<ref>) that can be written in the form 

(we use here the determinantal property ψ_2^2-ψ_1^2=1)

    ^-2ı g_c(κ)ıνψ_2^2(κ)-ℛ'(κ)A+ℛ(κ)-ℛ(κ)/κ+κA
    -ℛ^2(κ)+1/2κB+1+|ℛ(κ)|^2/κ-κB=ℛ(κ),
    
    ^-2ı g_c(κ)ıνψ_2^2(κ)-ℛ'(κ)B+ℛ(κ)-ℛ(κ)/κ+κB
    +ℛ^2(κ)+1/2κA-1+|ℛ(κ)|^2/κ-κA=1.

The system  of equations (<ref>) for A,B is clearly a system of four  linear equations for the four real variables A=A_1+ı A_2, B=B_1+ı B_2.
Introducing  the variables Z=A+ı B and W=A-ı B the system of equations (<ref>) can be recast in the form


    ^-2ı g_c(κ)ıνψ_2^2(κ)-ℛ'(κ)+ıℛ^2(κ)+1/2κZ+
    (ℛ(κ)-ℛ(κ)/κ+κ
    -ı1+|ℛ(κ)|^2/κ-κ)W=ℛ(κ)+ı ,
    
    (ı1+|ℛ(κ)|^2/κ-κ-ℛ(κ)-ℛ(κ)/κ+κ)Z
    +^-2ı g_c(κ)ıνψ_2^2(κ)-ℛ'(κ) W
    +ıℛ^2(κ)+1/2κW=R(κ)+ı.

Defining the quantities 

    E=^-2ı g_c(κ)ıνψ_2^2(κ)-ℛ'(κ),
       F=ℛ^2(κ)+1/2κ,
       H=ℛ(κ)-ℛ(κ)/κ+κ,
       G=1+|ℛ(κ)|^2/κ-κ,

the solution of (<ref>) is obtained as

    Z=A+ı B=|[   ℛ+ı H-ı G;   ℛ+ı  E+ıF ]||[  E+ı F  H-ı G; -H+ı G E+ı  F ]|=:r+ı s/f+ı h=f r+ s h /f^2+h^2+ıs f-r h/f^2+h^2,

and  similarly W=A-ı B=f r+s h /f^2+h^2-ıs f-r h/f^2+h^2,
where 

    r=ℛE-ℛH-F-G,
       s=E-H+ℛF+ℛG,
        f=|E|^2-|F|^2-G^2+H^2,
       h=2EF-HG.

Let us notice  that f,h are real, while r,s are complex-valued functions.

 The solution to MKdV is given by the formula

    q_breather=c+4(Aℛ-B)=c+4fℛ_2 r_1+ℛ_1 r_2-s_2+h(ℛ_2 s_1+ℛ_1 s_2+
    r_2)/f^2+h^2,

where we denote r=r_1+ı r_2 with r_1,r_2∈ and similarly for the other quantities.





To obtain the expression for f  defined  in (<ref>) we first observe that

    H=ℛ(κ)-ℛ(κ)/κ+κ=ıκ_1-χ_1cκ_1=ıκ_1κ_2-χ_1κ_2cκ_1κ_2=
    ıχ_2-κ_2cχ_2,

where we use the identity κ_1κ_2=χ_1χ_2,
and

    G=1+|ℛ(κ)|^2/κ-κ=c^2+(κ_1-χ_1)^2+(κ_2-χ_2)^22ıκ_2c^2=
    2κ_1(κ_2^2+χ_1^2-κ_1χ_1-κ_2χ_2)2ıκ_1κ_2c^2
         =2κ_2χ_2+χ_1κ_1-κ_1^2-χ_2^22ıχ_2c^2=c^2-|κ-χ|^22ıχ_2 c^2,

where we use the identity χ_1^2+κ_2^2-c^2=χ_2^2+κ_1^2  repeatedly.
Plugging into (<ref>) the  expressions for E,F from (<ref>),  and G and H from (<ref>) and (<ref>) respectively,  we obtain

    f|ψ_2|^4= f|κ+χ|^24|χ|^2=|^-2ı g_c(κ)ıν-ı c/2χ^2|^2-1c^4|κ+χ|^24|χ|^2|κ-χ|^2|+
             +1c^4χ_2^2(1/4(|κ-χ|^2-c^2)^2-c^2(κ_2-χ_2)^2)|κ+χ|^24|χ|^2
       =^4  g_c(κ))/|ν|^2-ıc/2^2  g_c(κ)(1-ı ν̅χ^2^2 ı g_c(κ)-1ı νχ^2^-2 ı g_c(κ))+χ_1^2(|χ|^2-c^2)4χ^4χ_2^2,

where we use the relation (κ+χ)(κ-χ)=-c^2 and 

    g_c(κ)=χ_1(4(χ_1^2-3χ_2^2-3/2c^2)t+x),
        g_c(κ)=χ_2(4(3χ_1^2-χ_2^2-3/2c^2)t+x).

In a similar way,

    h|ψ_2|^4   =^2 g_c(κ)( ^-2 ı g_c(κ) 2ıνχ+^2 ı g_c(κ)-2ı νχ)-c χ_22|χ|^4+c2|χ|^2χ_2
       =^2 g_c(κ)( ^-2 ı g_c(κ) 2ıνχ+^2 ı g_c(κ)-2ı νχ)+2c χ_1^22|χ|^4χ_2.

Long and involved algebraic manipulations  give the identities

    ( h_x-4χ_2h)|ψ_2|^4=2(ℛ_2 r_1+ℛ_1 r_2-s_2)|ψ_2|^4

and 

    ( f_x-4χ_2 f)|ψ_2|^4=-2(ℛ_2 s_1+ℛ_1 s_2+r_2)|ψ_2|^4.

Combining the above two expressions with (<ref>), (<ref>) and (<ref>) we can write the breather solution on a constant background in the form

    q_breather(x,t)   =c+2fh_x-f_x ff^2+h^2=c+2∂_xarctanh/f,

with f and h as in (<ref>) and (<ref>),  which coincides with the statement of the theorem.



 §.§ Model problem for the periodic travelling wave solution: the elliptic case
 
We consider a model problem that can be solved via elliptic functions. This model problem was first solved in <cit.> in the context of asymptotic analysis  for orthogonal polynomials related to Hermitian matrix models. The same problem appeared  in the long-time asymptotic analysis of the MKdV solution 
with step initial data when c_+=0<cit.>.
We introduce two real constant parameters c>d>0. The RH problem is as follows.

To find a 2× 2 matrix  M=M(x,t,;k) such that

  * M(x,t;k) is analytic in k∈ℂ∖[ıc,-ıc],
 
  * M_-(k)=M_+(k)J(k),
J(k)=[ 0 ı; ı 0 ],   k∈(ıc,ıd)∪(-ıd,-ıc),

^ı( x U+t V+ Δ)σ_3,   k∈(ıd,-ıd),

where  Δ is a real constant and 

    U=-πcK(m),   V=-2(c^2+d^2)U,   m=dc ,

where K(m) is the complete elliptic integral of the first kind;

  * M(k)→ I  as k→∞.

It follows from the standard scheme introduced by Zakharov, Shabat <cit.>, that 

    q(x,t) = 2ılim_k→∞kM_21(x,t;k) = 2ılim_k→∞kM_12(x,t;k)

satisfies MKdV equation (<ref>). 
The  explicit formula for M(k) from <cit.>, <cit.> is   constructed as follows.
We introduce the normalized holomorphic differential

    ω=-c4ı K(m)d z√((z^2+ d^2)(z^2+c^2)),   2∫_ıd̃^-ıd̃ω=1 ,

with K(m) as in (<ref>). 

For our purpose we fix the function √((z^2+ d^2)(z^2+c^2)) by   the condition that it is analytic off the intervals (ıc,ıd)∪(-ıd,-ıc) and positive at z=0. The intervals (ıc,ıd) and (-ıd,-ıc)  are oriented downwards.
We define the  β-cycle the 
counterclockwise loop encircling (ıc,ıd) and 
the α-cycle the path starting on the cut (ıc,ıd) on the left, going to the cut (-ıd,-ıc)  on the left and passing to the second sheet and reaching the cut (ıc,ıd) from the right on the second sheet.


We define the quantity

    τ=∫_βω.

It follows that

    τ=ı2K'(m)K(m),   m=dc,

where  K'(m)=K(√(1-m^2)) and K(m)=∫_0^π/2ds/√(1-m^2sin^2s).
Using  the relations <cit.>,  the quantity τ can also be written in the form
 
    τ=ı K'(m) K(m),    K(m)=(1+m)K(m),  m^2=4m(1+m)^2.

 Let us introduce the Jacobi theta-function with modulus τ
    θ(ζ)≡θ(ζ,τ)=∑_m=-∞^∞exp{πıτ m^2+2πıζ m}.

It is an even function of ζ and it has the following periodicity properties

    θ(ζ+1)=θ(ζ),  θ(ζ+τ)=^-ıπτ-2πıζθ(ζ).
 

Next we introduce the Abel map with base point ıc
    A(k):=∫_ıc^kω,

where ω is the holomorphic differential (<ref>)
and we observe that A(ı∞)=1/4.
Let us also introduce  the  function
 γ(k)=√(k-ıc/k-ıd)√(k+ıd/k+ıc), which is analytic off the intervals [ıc,ıd]∪[-ıd, -ıc], and tends to 1 at infinity.

Then the solution for the RH  problem <ref>  is given by <cit.>
    M(k)=θ(0)2θ(Ω)×
       ×[   (γ(k)+γ^-1(k))θ(A(k)-Ω-1/4)θ(A(k)-1/4) (γ(k)-γ^-1(k))θ(-A(k)-1/4-Ω)θ(-A(k)-1/4); ;   (γ(k)-γ^-1(k))θ(A(k)+1/4-Ω)θ(A(k)+1/4) (γ(k)+γ^-1(k))θ(-A(k)+1/4-Ω)θ(-A(k)+1/4) ],

where 

    Ω= xU+tV+Δ2π.

Using the expression for M(k) above, it  follows from (<ref>) that  the MKdV solution q(x,t) is given by

    q(x,t) = (c-d) θ(Ω+1/2; τ)/θ( Ω; τ)θ(0;τ)θ(1/2;τ).

We recall  the relation between the Jacobi θ-function and the elliptic function dn, <cit.>,
namely

    dn(2 K(m)z|m)=θ(1/2;τ)θ(0;τ)θ(z;τ)θ(z+1/2;τ)

with m as in (<ref>) and τ as in (<ref>).


Using the  above identities, the solution (<ref>) can be written in the form

    q(x,t) = (c+d)dn2K(m)(Ω+1/2) |  m.

According to <cit.>, formulas (162.01), p.38, and (165.05), p.41,

    dnu(1+m)| m
    =1-m sn^2(u|m)/1+m sn^2(u|m),   K(m)=K(m)(1+m),

with m  as in (<ref>).

Hence,  the expression  (<ref>) can be rewritten as 






    q(x,t)   =(c+d)1-d/c sn^22K( m) Ω+K(m)|m/1+d/c sn^22K( m)Ω+K(m)|m
       =-c-d+2c(c+d)c+d-d cn^2c(x-2(c^2+d^2)t)-ΔπK(m)+K(m) |m,

where to obtain the second expression we use the relation cn^2(u|m)+ sn^2(u|m)=1  and the explicit expressions of U and V as in (<ref>).
The second expression in (<ref>)   coincides with the travelling wave solution (<ref>) identifying β_3=c, β_2=d and β_1=0
and x_0=-ΔπK(m)+K(m) (see also Appendix <ref>).








 §.§ Model problem for the periodic  travelling wave solution: the hyperelliptic case

The model problem we are considering below   is obtained from the longtime asymptotic analysis of the MKdV RH problem in the oscillatory region when the step c_+>0.
It can be solved using hyperelliptic theta-functions.  The goal of this section is to show that such model problem still gives the periodic travelling 
wave solution  (<ref>) of the MKdV equation.  To reach our goal we introduce a conformal transformation  of the complex plane 
 and an auxiliary RH problem that is  going to reduce the hyperelliptic RH to an elliptic RH problem.  

 Find a 2× 2 matrix-valued  function W(k) analytic in \ [ı c_-,-ı c_-] such that
 
  * W_-(k)=W_+(k)J_W(k),   k∈ (ı c_-,-ı c_-)
with 

    J_W(x,t;k)=[ 0 ı; ı 0 ]    k∈(ı c_-,ı d)∪ (ı c_+,-ı c_+)∪(-ı d, -ı c_-)
       J_W(x,t;k)=[  ^ı (xU+t V + Δ)                0;                0 ^-ı (xU+t V + Δ) ],   k∈(ı d,ı c_+)∪ (-ı c_+,-ı d);

where c_->d>c_+,   and  

    U=-π√(c_ł^2-c_^2)K(m),     V=-2(c_ł^2+c_^2+d^2)U,

with  m^2=d^2-c_+^2c_-^2-c_+^2   and Δ  real constant.

  * W(k)= I+O(1/k)   k→∞;

  * W(k)  has at most fourth root singularities at the  points ±ı c_-, ±ı c_+ and ±ı d.

Then the quantity

    q_hel(x,t)=lim_k→∞2ikW_12(k;x,t)=lim_k→∞2ikW_21(k;x,t)

is a solution of the  MKdV  equation.

The   solution of the MKdV equation (<ref>)  obtained from the RH problem  <ref>   is the travelling wave solution (<ref>),  namely 

    q_hel(x,t)   =q_per(x,t, c_+,d,c_-,x_0)
       =-c_–d-c_++2(c_-+d)( c_++ c_-)/d+c_–(d-c_+)cn^2√(c_-^2-c_+^2) (x-𝒱t)+x_0|m,

with 𝒱=2(c_-^2+c_+^2+d^2),
and the phase x_0 takes the form

    x_0=-K(m)Δπ+K(m).

Here   cn(u |m) is the Jacobi elliptic function of modulus m^2=d^2-c_+^2c_-^2-c_+^2 and K(m) is the complete elliptic integral of the first kind of modulus m.







The RH  problem <ref>    has been considered in <cit.> where it was solved in terms of hyperelliptic theta-function defined on the Jacobi variety of the surface 
Γ:={(k,y)∈^2 |  y^2=(k^2+d^2)(k^2+c_-^2)(k^2+c_+^2)}. Such a surface has two automorphisms τ_1:(y,k)→ (y,-k)  and τ_2:(y,k)→ (-y,k).
Therefore the curve Γ covers two elliptic curves Γ_+:=Γ/τ_1 and Γ_-=Γ/(τ_1τ_2).
The corresponding genus 2 theta-function can be factorized as a product of Jacobi theta-function. However, pursuing this strategy, we did not see a simple way at arrive 
to the travelling wave solution (<ref>).
For this reason, we change  our  strategy and we formulate an auxiliary RH-problem that produces the desired solution and we connect such a  problem to our RH problem <ref>.


  §.§.§ Auxiliary RH  problem


We consider the  two real numbers c>d>0  with c^2=c_-^2-c_+^2 and d^2=d^2-c_+^2 and construct the following RH  problem for a 2× 2 matrix M_el=M_el(λ): 

  * M_el(λ) is analytic in λ∈ℂ∖[ıc,-ıc],
  * M_el,-(λ)=M_el,+(λ)J_el(λ), where

    J_el(λ)=[ 0 ı; ı 0 ],  λ∈(ıc,ıd)∪(-ıd,-ıc),

    J_el(λ)=^(ı xU+ı tV +ıΔ-Δ_4)σ_3,  λ∈(ıd,-ıd),

where Δ_4 is a constant to be determined and U, V and Δ, as in the RH  problem  <ref>.

  * M_el(λ)= I+O(1/λ) as λ→∞.
The  explicit formula for M_el(λ)   can be obtained from the solution of the RH problem <ref> as in (<ref>)  with Ω= 1/2π(xU+ tV +Δ+ıΔ_4). 











To proceed further, we make a transformation of the complex plane to reduce  the  RH  problem  for M_el(λ)  to the one in RH problem <ref>. 
We introduce a change of variable λ→ k defined as 

    λ=√(k^2+c_^2),     and denote   
    c_ł=√(c^2+c_^2),   d=√(d^2+c_^2).
  
The function λ=λ(k) is analytic for k∈ℂ\ [ı c_+,-ı c_+].
Next we introduce the matrix 

    Λ(k)=12[      a(k)+1a(k) -ı (a(k)-1a(k)); ı( a(k)-1a(k) )      a(k)+1a(k) ] M_el(λ(k)),
 where

    a(k)=√(k^2+c_^2/k^2).

Then the matrix Λ(k) is analytic for k∈ℂ\ [ı c_-,-ı c_- ] and satisfies the following conditions:
  
    Λ_-(k)=Λ_+(k)J_Λ(k),   k∈[ı c_ł,-ı c_ł]

  with 

    J_Λ(k)={[            [  0  1; -1  0 ],                  k∈(ı c_,0),;            [  0 -1;  1  0 ],                 k∈(0,-ı c_),;                [ 0 ı; ı 0 ], k∈(ı c_ł,ı d)∪(-ı d,-ı c_ł),;        ^ı (xU+tV+Δ+ıΔ_4)σ_3,   k∈(ı d,ı c_)∪(-ı c_,-ı d).                              ].

This is not exactly the RH problem <ref>  with jumps as in  <ref>.  We need to do some extra work.
For the purpose  we introduce  a scalar function F=F(k) analytic in k∈ℂ∖[ı c_ł,-ı c_ł], which satisfies the following conditions:

    F_+(k)F_-(k)=1,   k∈(ı c_ł,ı d)∪ (-ı d, -ı c_ł),

    F_+(k)F_-(k)=ı,   k∈(ı c_,0),     F_+(k)F_-(k)=-ı,   k∈(0,-ı c_),

    F_+(k)/F_-(k)=^Δ_4,   k∈(ı d,ı c_)∪(-ı c_,-ı d).


The quantity Δ_4   is independent from k and it has to be chosen in such a way that  F(k) is  bounded as k→∞.
The function F(k) can be represented in the following way:

    F(k)=exp   {R(k)/2πı∫_ı c_^0πı/2ṣ/(s-k)R_+(s)
    - ∫^-ı c__0πı/2ṣ/(s-k)R_+(s) +..
       +. .∫_ı d^ı c_Δ_4ṣ/(s-k)R(s) + ∫^-ı d_-ı c_Δ_4ṣ/(s-k)R(s)},

where R(k)=√((k^2+c_+^2)(k^2+c_-^2)(k^2+d^2)). The function R(k) is analytic for k∈\{[ı c_-,ı d]∪[ı c_+,-ı c_+]∪[-ı d,-ı c_-]} and positive and real at k=+0.  
The function F(k) is bounded at infinity provided that 
 
    Δ_4 =
    -πı/2∫_ı c_^0sṣ/R_+(s)/∫^ı c__ı dsṣ/R(s)=-2πı∫^c_+ _0ω,

where the one-form ω has been defined in (<ref>).
Hence 
    F(k)=1+𝒪(k^-1)   as    k→∞.

 Denote Λ^(1)(k)=Λ(k)
F^-σ_3(k).  The  jump conditions for the matrix  Λ^(1) are as follows:  
 Λ^(1)_-(k)=Λ^(1)_+(k)J^(1)(k)  with 

    J^(1)(k)={[                              [ 0 ı; ı 0 ], k∈(ı c_ł,ı d)∪(ı c_,-ı c_)∪(-ı d, -ı c_ł),;                           ^ı (xU+tV+Δ)σ_3,                 k∈(ı d,ı c_)∪(-ı c_,-ı d).                                            ].


The matrix Λ^(1)(k) is not exactly  the solution of the hyperelliptic model problem <ref>, since it has poles at the point  k=0.
This is because the function F(k) is vanishing as √(k) as k→0 with k>0, and is growing as 1/√(k) as k→0 with k<0.
Hence, the second column of Λ^(1) has a pole of the  first order when k→ 0 with k<0, and the first column of Λ^(1) has a pole when k→0 with k>0.




Direct analysis of  the  behavior of Λ^(1)(k) at k→0 shows
that the matrix  function

    Λ^(2)(k):=[ 1+α/k  ıα/k;  ıα/k 1-α/k ]Λ^(1)(k)

does not have pole at k=0 provided that

    α=-c_/2M_el,11(c_)-ı M_el,21(c_)/M_el,11(c_)+ı M_el,21(c_),

 where M_el(k) is as in (<ref>) by taking care of replacing Δ by Δ+ıΔ_4. 
We arrive at  the following lemma.

The 2×2  matrix  Λ^(2)(k)  defined in (<ref>)  with α as in (<ref>)  is the unique solution to the
hyperelliptic model problem in (<ref>).

Further, the solution of the MKdV equation is given by the
 formula:

    q_hel(x,t)=lim_k→∞2ı kΛ^(2)_21(k)=2ılim_k→∞kM_el21(k)-2α=
    2ılim_k→∞kM_el12(k)-2α,

so that, plugging into the above expression the explicit expression of M_ell(k)  and α we obtain 

    q_hel(x,t)=(c-d)θ(1/2+Ω-Δ_4/2πı;τ)/θ(Ω-Δ_4/2πı;τ)θ(0;τ)θ(1/2;τ)+c_M_el,11(c_)-ı M_el,21(c_)/M_el,11(c_)+ı M_el,21(c_),

with τ=ı2K'(m)K(m),m=dc, and

    Ω=xU+tV+Δ2π,

 with U, V and Δ as in the  RH problem with jumps as in (<ref>).
Summarizing we have obtained the solution of the hyperelliptic RH problem  with jumps as in (<ref>)   and therefore of the MKdV equation
in terms of elliptic functions.  We need to do some extra work to show that the expression (<ref>)  coincides with the travelling wave solution of the MKdV equation.


In order to prove Theorem <ref>, we need to show that the quantity q_hel(x,t)  in (<ref>) is equal to   the  travelling wave solution of the MKdV equation defined in (<ref>), namely we have to prove the relation

    q_hel(x,t)=-c_–d-c_++2(c_-+d)( c_++ c_-)/d+c_–(d-c_+)cn^2√(c_-^2-c_+^2) (x-𝒱t)+x_0|m,

where the phase x_0 takes the form

    x_0=K(m)Δπ+K(m),   m^2=d^2-c_+^2c_-^2-c_+^2.



For the purpose we need a series of identities among elliptic functions.
We first consider the term α in (<ref>). We  observe   from the relation (<ref>) and (<ref>) that 

    A(c_)=∫_ic^c_ω=-τ/2+1/4-Δ_42π i

so that the quantity α in (<ref>) takes the form
 
    α   =-c_/2(γ(c_)+γ^-1(c_))
     θ(-τ/2-Ω;τ)/θ(-τ/2-Δ_4/2πı;τ)-
     ı(γ(c_)-γ^-1(c_))θ(-τ/2+1/2-Ω;τ)/θ(-τ/2+1/2-Δ_4/2πı;τ)/(γ(c_)+γ^-1(c_))θ(-τ/2Ω;τ)/θ(-τ/2-Δ_4/2πı;τ)+
     ı (γ(c_)-γ^-1(c_))θ(-τ/2+1/2-Ω;τ)/θ(-τ/2+1/2-Δ_4/2πı;τ)
       =-c_/2  (γ(c_)+γ^-1(c_))θ(-τ/2-Ω;τ)/θ(-τ/2+1/2-Ω;τ)
    - ı (γ(c_)-γ^-1(c_)) θ(-τ/2-Δ_4/2πı;τ)/θ(-τ/2+1/2-Δ_4/2πı;τ)/(γ(c_)+γ^-1(c_)) θ(-τ/2-Ω;τ)/θ(-τ/2+1/2-Ω;τ)+
     ı (γ(c_)-γ^-1(c_)) θ(-τ/2-Δ_4/2πı;τ)/θ(-τ/2+1/2-Δ_4/2πı;τ).


 In order to simplify the above expression we use the  identities  <cit.>
    θ(A(k)-1/4;τ)/θ(A(k)+1/4;τ)
    =
    √(c+d/c-d)·γ(k)+γ^-1(k)/η(k)+η(k)^-1,
 
where
η(k)=√(k-ıc/k+ıc)√(k-ıd/k+ıd) and  use  the following periodicity property of elliptic functions:

    dn(u+iK'(m) |m)=-ıcn(u|m)sn(u|m),   K'=K(√(1-m^2)),

    dn^2(u |m)=1-m^2sn^2(u |m),  sn^2(u |m)+cn^2(u |m)=1,

where m is defined in (<ref>).
Using the above  three identities  and (<ref>) we arrive at the following form for q_hel(x,t) in (<ref>):

    q_hel(x,t)   =(d+c) dn2 K(m)(1/2+Ω-Δ_4/2πı)|m+
       +c_ı√(1-m1+m)sn(2 K(m)Ω+K(m)|m)cn(2 K(m)Ω+K(m)|m) -√(c_–dc_-+d)ı√(1-m1+m)sn(2 K(m)Ω+K(m)|m)cn(2 K(m)Ω+K(m)|m) +√(c_–dc_-+d) .

Next we use the addition formula for Jacobi elliptic function
 dn<cit.>
    dn(u+v|m)=dn(u|m) dn(v|m)-m^2 sn(u|m)cn(u|m)sn(v|m)cn(v|m)/1-m^2 sn^2(u|m)sn^2(v|m),

and the following relations  <cit.>
    sn(u|m) = (1+m)snu/1+m|m/1+m sn^2u/1+m|m,
    cn(u|m) = cnu/1+m|m dnu/1+m|m/1+m sn^2u/1+m|m,
    dn(u|m) = 1-m sn^2u/1+m|m/1+m sn^2u/1+m|m,
     m^2=4m/(1+m)^2,   K(m)=K(m)(1+m).

 We also obtain the  following relations 

    sn(ı K[m]Δ_4/π|m)=ı (c+d)√((d-d)(c_ł+c))/(c_ł+c-d+d)√(cd ),
       cn(ı K[m]Δ_4/π|m)=
    (c_łd+d c)√(c_ł+c)/√(cd(d+d) ) (c_ł+c-d+d),
    
       dn(ı K[m]Δ_4/π|m)=
    c_ł+c+d-d/c_ł+c-d+d=c-d/c_l-d=c_l+d/c+d,

and the further identity

    2(c_l+c)(c_ld+dc)/(c_l+c-d+d)(c_l+c+d-
    d)(d+d)=1.

Substituting the above relations in (<ref>)    and defining  Ω=2K(m)Ω+K(m), we obtain

    q_hel(x,t)   =1-m^2 sn^4(Ω| m)+c_+c^2(d+c_-) sn^2(Ω| m)-
    c_+ cn^2(Ω| m) dn^2(Ω| m)/d+c_-(d+c_-) c^2sn^2(Ω| m)+cn^2(Ω| m) dn^2(Ω| m)/d+c_-
       =-c_++(d+c_-)1-d-c_+/c_-+c_+sn^2(Ω| m)1+d-c_+/c_-+c_+sn^2(Ω| m)
       =-c_+-c_–d+2(c_-+d)(c_++c_-)c_-+d-(d-c_+)cn^2(Ω| m),

where 

    Ω=√(c_-^2-c_+^2)(x-2(c_-^2+c_+^2+d^2)t)+x_0,   
    x_0=-K(m)Δπ+K(m).

which concludes the proof of Theorem <ref>.


§ LARGE TIME ASYMPTOTICS: PROOF OF THEOREM <REF> PART (B)

We  study the long-time asymptotics of the RH  problem <ref> by  applying   the  Deift-Zhou steepest descend method <cit.> for oscillatory RH problems. The high oscillatory terms of  the matrix entries of J_M(k)  defined in (<ref>) come from the exponential  factors 
 ^±ıθ(x,t;k).
Since the stationary point of θ(x,t;k)=kx+4k^3t is k=√(-x/12t), we introduce  a new independent variable

    ξ=x12t,

and the function θ(k,ξ) with  tθ(k,ξ)=θ(x,t;k), namely

    θ(k,ξ)=12kξ+4k^3.

The sign of the θ(k,ξ) are plot in  Figure <ref>.



To perform the asymptotic analysis  of the Riemann Hilbert problem <ref>, our first step is the introduction of a scalar  function g=g(k,ξ)  
which is asymptotic to θ(k,ξ), namely

    g(k,ξ)=θ(k,ξ)+O(1/k),   |k|→∞.





The function g(k,ξ)  takes different forms for different regions of the parameter ξ (<cit.>):

    g(k,ξ)=ĝ_c_+(k,ξ)
      ξ≥c_-^2/3+c_+^2/6,
    
    
    ∫_ı c_-^k12 s(s^2+ξ+c_-^2+c_+^2-d^2/2)√(s^2+d(ξ)^2) ṣ/√((s^2+c_-^2)(s^2+c_+^2)),
      -c_-^2/2+c_+^2 ≤ξ≤c_-^2/3+c_+^2/6,
    
    
    ĝ_c_-(k,ξ),  ξ≤-c_-^2/2+c_+^2.

where the function  ĝ_c(k,ξ)=t^-1 g_c(x,t;k)=(4k^2-2c^2 + 12ξ)√(k^2+c^2), and  g_c(x,t;k) has been defined in (<ref>). 
The quantity  d(ξ)  appearing in the function g(k,ξ) in the middle region,  is a function of the parameter ξ,  and it is obtained from (<ref>) below and satisfies c_+<d(ξ)<c_-.

We observe that the function g(k,ξ) is  Schwartz symmetric. Indeed  ĝ_c(k̅,ξ)=ĝ_c(k,ξ), while in the middle region, we have  

    g(k,ξ)   =(∫_ı c_-^k+1/2∫_-ı c_-^ı c_-)12 s(s^2+ξ+c_-^2+c_+^2-d^2/2)√(s^2+d(ξ)^2) ṣ/√((s^2+c_-^2)(s^2+c_+^2))
       =1/2(∫_ı c_-^k+∫_-ı c_-^k)12 s(s^2+ξ+c_-^2+c_+^2-d^2/2)√(s^2+d(ξ)^2) ṣ/√((s^2+c_-^2)(s^2+c_+^2))

where the second integral on the first line is equal to zero because of   (<ref>) and   the residue theorem. Clearly the second line of the above expression is  Schwartz symmetric.
  Then we define the first transformation of the RH problem <ref>(note that since the large time asymptotics is studied along the rays x/t=const,  the parameters x,t in M are changed to parameters ξ=x/(12t),t in Y )
    Y(ξ,t;k)=M(x,t;k)^ıtg(k,ξ)-θ(x,t;k)σ_3,

so that

    Y_-(ξ,t;k)=Y_+(ξ,t;k)J_Y(ξ,t;k),    k∈Σ,

where
 
    J_Y(ξ,t;k)=^-ıtg_+(k,ξ)-θ(x,t;k)σ_3J_M(x,t;k)^ıtg_-(k,ξ)-θ(x,t;k)σ_3

 with J_M as in (<ref>).



The three different regions in the definition of g will be referred to as 
  *   a dispersive shock wave region -c_-^2/2+c_+^2 < ξ < c_-^2/3+c_+^2/6, that can contain breathers on an elliptic background;


  *   right constant region  ξ≥c_-^2/3+c_+^2/6, with possible solitons and breathers on the constant background c_+, travelling in positive direction;


  *  left constant region -c_-^2/2+c_+^2 ≤ξ≤c_-^2/3+c_+^2/6,  where possible breathers on the constant background c_- travel in either positive or negative direction.
 
 Furthermore, the left constant region is subdivided into two regions:
 
  *  utmost left constant region ξ<-c_-^2/2;
  *  middle left constant region -c_-^2/2<ξ<-c_-^2/2+c_+^2.































































































































































































































We start by performing the asymptotic analysis of the dispersive shock wave region.







 §.§ Proof of theorem <ref>  part (b): dispersive shock wave region with  -c_ł^2/2+c_^2<x/12 t<c_ł^2/3+c_^2/6

In this region we will verify a-posteriori that the g-function  is analytic in ℂ\ [ı c_-, -ı c_-] and  takes the form
 
    g(k,ξ)=12∫_ı c_-^ks(s^2+ ξ+c_-^2+c_+^2/2-d^2/2)√(s^2+d^2(ξ))ṣ/√((s^2+c_-^2)(s^2+c_+^2)) ,

 for
  -c_-^2/2+c_+^2<ξ<c_-^2/3+c_+^2/6,  where the quantity d=d(ξ)  is determined by 
  
    ∫_ı d^ı c_+s(s^2+ ξ+c_-^2+c_+^2/2-d^2/2)√(s^2+d^2)ṣ/√((s^2+c_-^2)(s^2+c_+^2))=0.
r4cm[scale=0.4]




[fill=black] (0,1) circle [radius=0.05];
[fill=black] (0,-1) circle [radius=0.05];
[fill=black] (0,2.5) circle [radius=0.05];
[fill=black] (0,-2.5) circle [radius=0.05];
[fill=black] (-1,1.5) circle [radius=0.1];
[fill=black] (-1,-1.5) circle [radius=0.1];
[fill=black] (1,-1.5) circle [radius=0.1];
[fill=black] (1,1.5) circle [radius=0.1];
[fill=red] (-2,2.5) circle [radius=0.1];
[fill=red] (-2,-2.5) circle [radius=0.1];
[fill=red] (2,-2.5) circle [radius=0.1];
[fill=red] (2,2.5) circle [radius=0.1];
[very thin] (0,-3.5) to (0,3.5);
[very thick] (-3,5) [out=-60, in=-180] to (0,2.5) [out=0,in=-120] to (3,5);
[very thick] (-3,-5) [out=60, in=180] to (0,-2.5) [out=0,in=120] to (3,-5);
[very thick] (-3,3) [out=-60, in=-180] to (0,1.) [out=0,in=-120] to (3,3);
[very thick] (-3,-3) [out=60, in=180] to (0,-1.) [out=0,in=120] to (3,-3);

 at (0.7,.5) ı c_+;
 at (0.7,3.1) ı c_-;
 at (0.85,-.7) -ı c_+;
 at (0.85,-2.3) -ı c_-;

[very thick, dashed] (-3,0) to (3,0);
The  zero set described by  (<ref>) and the  spectra  of two  distinct  breathers  trapped inside the dispersive shock wave region.
The solvability of this   equation  for d=d(ξ)  was established in <cit.>.  Below we give a different derivation of (<ref>) and we show the solvability  of d=d(ξ) using the hyperbolic
nature of the Whitham modulation equations.
The dispersive shock wave region contains a trapped breather if  the discrete spectrum of  the breather is contained within the region bounded by the curves

    g_c_+(k, c_-^2/3+c_+^2/6)=0,   g_c_-(k,-c_-^2/2+c_+^2),

where  12(c_-^2/3+c_+^2/6) and 12(-c_-^2/2+c_+^2) are the speeds   respectively of the leading edge and trailing edge of the dispersive shock wave (see Figure <ref>  for the level set of (<ref>)).
A plot of the signs of the g(k,ξ) is shown in Figure <ref>, which describes  the regions of the k-plane  where the quantity
^ı t g(k,ξ) is exponentially small.


Our first step in the asymptotic analysis is to take care of the discrete spectrum.
We introduce the function T(k,ξ) defined as

    T(k,ξ)=T(k)H(k),
       T(k)=∏_κ_j | g(κ_j,ξ)<0
    κ_j>0, κ_j>0(k-κ_jk-κ_jk+κ_jk+κ_j)
    ∏_κ_j  | g(κ_j,ξ)<0
    κ_j>0
    κ_j=0(k-κ_jk-κ_j)

where  H(k) is analytic in \ [ic_ł,-ic_ł]  and H(k)= 1+O(k^-1) as |k|→∞. Further properties  of T(k)   and H(k)  will be determined later.

 Then we define   the first transformation 


    Y(ξ,t;k)=Y(ξ,t;k)T^-σ_3(ξ,t;k)×
       ×[                                   1 -^-2ı t g(k,ξ)/[ıν_j/T^2(k)(k-κ_j)];    [ıν_j/T^2(k)(k-κ_j)]^2ı t g(k,ξ)                                   0 ], 
    |k-κ_j|<ε,    g(κ_j,ξ)≤-δ
    [                                    1 -^-2ı t g(k,ξ)/[ı ν_j/T^2(k)(k+κ_j)];    [ı ν_j/T^2(k)(k+κ_j)]^2ı t g(k,ξ)                                    0 ], 
    |k+κ_j|<ε,  g(-κ_j,ξ)≤-δ
    [                                   0 [ı ν_j T^2(k)/(k-κ_j)]^-2ı t g(k,ξ); ^2ı t g(k,ξ)[(k-κ_j)/-ı ν_j T^2(k)]                                   1 ],   |k-κ_j|<ε,  g(κ_j,ξ)≥δ,
    
    [                                  0  [ıν_jT^2(k)/(k+κ_j)]^-2ı t g(k,ξ); ^2ı t g(k,ξ)[(k+κ_j)/-ıν_j T^2(k)]                                  1 ],   |k+κ_j|<ε,  g(-κ_j,ξ)≥δ,
    
    
    I,  elsewhere.

We assume ε sufficiently small so that g(κ,ξ)<-δ/2 for |k-κ_j|≤ε and similarly for the other cases.
In this way we are taking control of the exponentially big terms in the jump matrix related to all points  of the discrete  spectrum except for those  κ_j  for which 
g(κ_j,ξ)=0,   for some ξ with -c_-^2/2+c_+^2 < ξ<c_-^2/3+c_+^2/6.  Following the   discussion in the introduction, this is possible only for points of the  spectrum corresponding to breathers.







 The jump matrix J_Y (  associated to the RH problem Y_- = Y_+ J_Y), is given by

 
    J_Y(ξ,t;k)=[                                 1 T^2(k,ξ)(k-κ_j)/ıν_j ^2ı t g(k,ξ);                                 0                                 1 ]  
    [                                  1                                  0; ıν_j ^2ı t g(k,ξ)/(k-κ_j) T^2(k,ξ)                                  1 ]  ,    k∈ C_j;
    
    [                                  1 T^2(k,ξ)(k+κ_j)/ı ν_j ^2ı t g(k,ξ);                                  0                                  1 ]  
    [                                   1                                   0; ı ν_j ^2ı t g(k,ξ)/(k+κ_j) T^2(k,ξ)                                   1 ]   g(k)>δ/2,  k∈ -C_j;
    
    [                                  1                                  0; (k-κ_j)^2ı t g(k,ξ)/ı ν_j T^2(k,ξ)                                  1 ]  
    [                                  1 ı ν_j T^2(k,ξ)/(k-κ_j)^2ı t g(k,ξ);                                  0                                  1 ]     k∈C_j;
    
    [                                1                                0; (k+κ_j)^2ı t g(k,ξ)/ıν_jT^2(k,ξ)                                1 ]  
    [                                   1 ı ν_j T^2(k,ξ)/(k+κ_j) ^2ı t g(k,ξ);                                   0                                   1 ]  ,    k∈-C_j;
    
    
    
    T_+^σ_3(k,ξ)J_Y(k)T_-^-σ_3(k,ξ),





 It is clear from the form of the above jumps, that the matrix J_Y will be exponentially close to the identity  as t→∞ on the circles ± C_j and ±C_j for all those   points  κ_j  of the discrete spectrum 
 for which g(κ_j,ξ)≠ 0 when ξ is in the region -c_-^2/2+c_+^2 < ξ<c_-^2/3+c_+^2/6. 

The next  step is to  take care of the continuous spectrum  on the real axis.  As a first step, we   reduce the jump J_Y(t,ξ,k)  for k∈ℝ\{0} to a matrix exponentially close to the identity.
   For the purpose it  is  sufficient to  factorise  the matrix J_Y(t,ξ,k) to the form
  
    J_Y(ξ,t;k)   =[                      1 -r(k)T^2(k)^-2itg(k,ξ); -r(k)T^2(k)^2ı tg(k,ξ)             1+|r(k)|^2 ],   k∈ℝ∖{0}
       =[                      1                      0; -r(k)T^2(k)^2ı tg(k,ξ)                      1 ][                      1 -r(k)T^2(k)^-2itg(k,ξ);                      0                      1 ].


Then  using Deift-Zhou contour deformation  method  we introduce the new matrix X,
    X(ξ,t;k)={[                                                                                                                     Y(ξ,t;k)[                      1                      0; -r(k)T^2(k)^2ı tg(k,ξ)                      1 ],    k∈Ω_1,;                                                                                                                    ;                                                                                                                         Y(ξ,t;k)
    [                     1 r(k)T^2(k)^-2itg(k,ξ);                     0                     1 ],   k∈Ω_2,;                                                                                                                                                                                                                               Y(ξ,t;k), ].

where the regions Ω_1 and Ω_2 are specified   in Figure <ref> and do not contain points of the discrete spectrum.
In this way  we can reduce the jump of X(k)  on  ℝ\{0}  to identity, while the jumps of X(k) on the lines L_1,2\ U_±ı d
 are exponentially close to the identity, where U_±ı d is a small neighbourhood of the point ±ı d.  The jumps J_X(k) for  k∈ L_1,2∩ U_±ı d give the subleading  contribution to the asymptotics, analysis.  

 We still need to determine the functions g(k,ξ) and F(k).
  The remaining jumps of the matrix X(k) are  obtained using the identities  from Lemma <ref>, and take the following  form:

    J_X(ξ,t;k)=
    T^σ_3_+(k)[    ^-ı t(g_+-g_-)                 0; f(k)^ı t(g_++g_-)     ^ı t(g_+-g_-) ]T^-σ_3_-(k)    k∈(ı c_ł,ı d),
    
    T^σ_3_+(k) ^-ı t (g_+-g_-)σ_3 T^-σ_3_-(k),   k∈(id,ic_)∪ (-ic_,-id),
    
    T^σ_3_+(k)[               0 ı^-ı t(g_++g_-);  ı^ı t(g_++g_-)               0 ]T^-σ_3_-(k),   k∈(ı c_,-ic_),
    
    
    T^σ_3_+(k)[      ^-ı t(g_+-g_-) -f(k)^-ı t(g_++g_-);                   0       ^ı t(g_+-g_-) ]T^-σ_3_-(k),k∈(-ı d,-ı c_ł).

  
We require that the above matrix J_X  has oscillatory   diagonal terms and non oscillatory  off-diagonal terms as t→∞.
Therefore we need to  require  that 

    g_+(k)+g_-(k)=0,   k∈[ı c_-,ı d]∪[ı c_+,-ı c_+]∪ [-ı d,-i c_-], 
       g_+(k)-g_-(k)∈,   k∈[ı c_-,-ı c_-],
        g_+(k)-g_-(k)=-B(ξ),   k∈ (ı d,ı c_+)∪ (-ı c_+,-ı d),

where B=B(ξ) is independent from k.  Furthermore, for reasons that will become clear later,  we chose the scalar function T(k) in such a way that

    T_+(k)T_-(k)=1,   k∈ [ı  c_+,-ı c_+].

Then the above jump matrices are reduced to the form

    J_X(ξ,t;k)={[                                                                                                                                               [ T_+(k)^-ı t (g_+-g_-)/T_-(k)                            0;             f(k)T_-(k)T_+(k)   T_-(k)^ı t(g_+-g_-)/T_+(k) ],    k∈(ı c_-,ı d),;                                                                                                                                                       [  T_+(k)^ı t B(ξ)/T_-(k)                       0;                       0 T_-(k)^-ı t B(ξ)/T_+(k) ],   k∈(ı d,ı c_+)∪ (-ı c_,-ı d);                                                                                                                                                                                                                                                            [ 0 ı; ı 0 ],   k∈(ı c_,-ic_),;                                                                                                                                              [ T_+(k)^-ı t (g_+-g_-)/T_-(k)            -f(k)T_-(k)T_+(k);                            0   T_-(k)^ı t(g_+-g_-)/T_+(k) ],   k∈(-ı d,-ı c_ł).;                                                                                                                                              ].


  §.§.§ Determination of   the scalar functions T(k)  and g(k).

In this subsection we determine the scalar function T(k) and we derive the expression for the function g(k)  that satisfies (<ref>) and (<ref>).

The function T(k)  satisfies the relation (<ref>) and  (<ref>). We still need to add some assumptions on the boundary values of T(k) in the interval
(ı d,ı c_+) and (-ı c_+,-ı d). In order to obtain a constant jump  matrix J_X  in (<ref>)  we assume that T_+(k)=T_-(k)^ıΔ for k∈(ı d,ı c_+)∪ (-ı c_+,-ı d), where the function Δ=Δ(ξ)   will be independent from k and needs to be determined.
Using the relations (<ref>)  we   finally have  the following RH problem for the function T(k):
    T_+(k)T_-(k)=1|a(k)|^2   k∈ (ı c_ł,ı d),
        T_+(k)T_-(k)=|a(k)|^2   k∈∪ (-ı d,-ı c_ł),
       T_+(k)=T_-(k)^ıΔ(k),   k∈(ı d,ı c_)∪ (-ı c_,-ı d),
       T_+(k)T_-(k)=1,   k∈ (ı  c_,-ı c_),
       T(k)=1+O(k^-1),   |k|→∞.

The solution is obtained passing to the logarithm and using the Plemelj formula. Let us introduce

    R(k)=√((k^2+c_ł^2)(k^2+d^2)(k^2+c_^2)),

where R(k) is analytic in \ [ı c_ł,ı d]∪[ı c_,-ı c_]∪ [-ı d,-i c_ł]   and real positive for k=+0, where +0 means the limit to 0 from the right. Then the expression

    T(k)=T(k)
    exp[R(k)/2πı{( ∫_ı c_ł^ı d+∫^-ı c_ł_-ı d+∫^-ı c__ı c_)-logT^2(s)(s-k)R_+(s)ds..
       +..( ∫^-ı c_ł_-ı d-∫_ı c_ł^ı d)ln|a(s)|^2/(s-k)R_+(s)ds+
    (∫_ı d^ı c_+∫_-ı c_^-ı d)ıΔṣ/(s-k)R(s)}]

solves the scalar RH problem for T(k) with the quantity T(k) defined in (<ref>).
The requirement that T(k)=1+O(k^-1) as |k|→∞ determines Δ(ξ). Indeed we have, using the symmetry of the problem, that

    Δ   = ∫^ı c_ł_ı dln(|a(s)|^2T^2(s))sṣ/R_+(s)+∫^ı c__0lnT^2(s)sṣ/R_+(s)/-ı∫_ı d^ı c_sṣ/R(s)
       =-√(c_-^2-c_+^2)K(m)[∫^ı c_ł_ı dln(|a(s)|^2T^2(s))sṣ/R_+(s)+∫^ı c__0lnT^2(s)sṣ/R_+(s)].



The scalar function g(k) satisfies the conditions (<ref>) and (<ref>). This implies that the function g'(k)  is analytic in \ [ı c_ł,-ı c_ł] and on the interval [ı c_ł,-ı c_ł] satisfies the conditions

    g'_+(k)+g'_-(k)=0,   k∈(ı c_ł,ı d)∪(ı c_,-ı c_)∪ (-ı d,-i c_ł)
       g'_+(k)-g'_-(k)=0,   k∈ (ı d,ı c_)∪ (-ı c_,-ı d)
       g'(k)=θ̂'(k)+O(k^-2)  .

From the above conditions it follows that 

    g'(k)=12P(k) R(k),   P(k)=k^5+k^3(ξ+1/2(d^2+c_ł^2+c_^2))+bk

where R(k) is  defined in (<ref>). The constant b  in (<ref>)  is determined by  requiring that the integral

    g(k)=∫_ı c_ł^kg'(s)ds

satisfies the third relation in (<ref>). This   immediately  implies that 

    ∫^ı c__ı dg'(k)dk=0,  ∫^-ı c__-ı dg'(k)dk=0

which gives 
    b=13(c_ł^2c_^2+c_ł^2d^2+c_^2d^2)+(ξ-1/6(c_ł^2+c_^2+d^2))[c_ł^2-(c_ł^2-c_^2)E(m)K(m)],

where E(m)=∫_0^π/2√(1-m^2sin^2θ)dθ and K(m)=∫_0^π/2dθ/√(1-m^2sin^2θ) are the complete elliptic integrals of the second and first kind respectively with modulus m^2=d^2-c_^2c_ł^2-c_^2.
 We also have

    B(ξ)   =2∫^ı c_ł_ı dg_+'(k)dk=12∫_-d^2^-c^2_-s^2+s(ξ+1/2(d^2+c_ł^2+c_^2))+b√((s+d^2)(s+c_ł^2)(s+c_^2))ds
       =-12π√(c_ł^2-c_^2)K(m)(ξ-16(c_ł^2+c_^2+d^2))∈.

Let us observe that

    tB(ξ)=xU+tV

where U and V have been defined in (<ref>).
 We still need to determine the  quantity d. This is obtained by requiring that g'(k)|_k=±ı d=0 that implies that the polynomial P(k) in (<ref>)  has a zero at k=±ı d, namely
 
    P(±ı d)=±ı[d^5-d^3(ξ+1/2(d^2+c_ł^2+c_^2))+bd]=0
       12 ξ=W_2(c_,d,c_ł) ,

 where W_2(β_1,β_2,β_3), β_1≤β_2≤β_3 has been defined in (<ref>).
 
 We observe that W_2(β_1,β_2,β_3) is the speed of the Whitham modulation equations for MKdV derived in <cit.>.
 The relation with the speed  V_2(r_1,r_2,r_3)  of the Whitham modulation equations for KdV  <cit.> is as follows:
 
    W_2(β_1,β_2,β_3)=V_2(β^2_1,β^2_2,β^2_3) .

 In particular it was shown in <cit.> that the Whitham modulation equations for KdV are strictly hyperbolic and  satisfy the relation ∂_r_2V_2(r_1,r_2,r_3)>0 for r_1<r_2<r_3 which 
 implies that
 
    ∂∂β_2W_2(β_1,β_2,β_3)=2β_2 ∂∂ r_2V_2(β_1^2,r_2,β_3^2)|_r_2=β_2^2>0,  β_2>0.

 The above relation shows that the equation (<ref>) is invertible for d as a function of ξ only when d>0 or equivalently when c_+>0.
  Further comments about the case c_->-c_+>0 are given in Appendix <ref>.
 
 Using the properties of the elliptic functions <cit.>, we get that as m→ 0
    K(m)=π2(1+m^24+964m^4+O(m^6)), 
    E(m)=π2(1-m^24-364m^4+O(m^6)),

and as m→ 1
    E(m)= 1+12(1-m)[log161-m^2-1](1+o(1)),
       K(m)= 12log161-m^2(1+o(1)).

Using the above expansions  we have that
 
  *  as m→ 0 or d→ c_+,
    W_2(c_,c_,c_ł)=-6c_ł^2+12c_^2

 and 
 
  *   as m→ 1 or d→ c_ł,
    W_2(c_,c_ł,c_ł)=4c_ł^2+2c_^2

 which implies that 
 
    -c_ł^2/2+c_^2<ξ<c_ł^2/3+c_^2/6.

 
 Summarizing,  the function g(k)=g(k,ξ) takes the form
 
    g(k)   =12∫_ı c_-^ks^5+s^3(ξ+1/2(d^2+c_ł^2+c_^2))+bs √((s^2+c_-^2)(s^2+c_+^2)(s^2+d^2))ds,
    
      b   =13(c_ł^2c_^2+c_ł^2d^2+c_^2d^2)+(ξ-1/6(c_ł^2+c_^2+d^2))[c_ł^2-(c_ł^2-c_^2)E(m)K(m)] .

 The parameter d=d(ξ) is determined by (<ref>). The above 
 derivation  for the function g(k)  is equivalent to the one obtained in <cit.> and written in (<ref>)  with d=d(ξ) as in (<ref>).
 The signature of g(k) is given in Figure <ref>.
 
 
 
 Signature of g(k) can be constructed from the following considerations.
 Let us look at a level line g(k) = const. It can be parametrised  as  k = k(s), where s∈ℝ, with |k'(s)| = 1.  The function g(k(s)) equals
 
    g(k(s)) = g(k(s_0))+∫^s_s_0 g'(k(σ)) k'(σ) σ̣.
 
 Since g(k(s)) = const, hence 
 
    k'(s) =  g'(k(s))|g'(k(s))|.

 This immediately gives us the direction of the level line for every point k, and there is  exactly one line passing through every such a point  as long as  g'(k)≠ 0.
 Except for those regular points, we have several singular points, where g'(k) is either 0 or infinite.
  For example, near the point k=±ı d one has  g(k)=∫ ^ı d_ı c_-g'(s)ds+∫_ı d^kg'(s)ds  so that 
 
    g(k)∼2/3(k-ı d)^3/2e^ıπ/4×,

 where the constant is real. So the lines where g(k)=0 coming out of the point k=ı d have argument φ=-π/6+2/3nπ with n∈ℤ.  This corresponds to three different lines  with angles π/2,-π/6,7π/6.
 In a similar way there are three lines emerging from the point k=-ı d.











Also,  there are 6 rays g = const,  converging to the point k→∞,  along directions π k/3,k =0,1,2,3,4,5.
These rays consist of the real axis and the two ray emerging from the points ±ı d (see  Figure <ref>).




 

  §.§.§ Opening of the lenses

With our choice of the functions T(k) and g(k,ξ), the jump J_X in (<ref>)  reduces to the form

    J_X(ξ,t;k)={[                                                                                                                                                                       [ T_+(k)^-ı t (g_+(k)-g_-(k))/T_-(k)                                  0;                   f(k)T_-(k)T_+(k)   T_-(k)^ı t(g_+(k)-g_-(k))/T_+(k) ],    k∈(ı c_-,ı d),;                                                                                                                                                                     ;                                                                                                                                                                                                                                                                                      e^ı (t B(ξ)+Δ(ξ))σ_3,   k∈(ı d,ı c_+)∪ (-ı c_,-ı d),;                                                                                                                                                                     ;                                                                                                                                                                                                                                                                                                            [ 0 ı; ı 0 ],   k∈(ı c_,-ic_),;                                                                                                                                                                     ;                                                                                                                                                                      [ T_+(k)^-ı t (g_+(k)-g_-(k))/T_-(k)                  -f(k)T_-(k)T_+(k);                                  0   T_-(k)^ı t(g_+(k)-g_-(k))/T_+(k) ],   k∈(-ı d,-ı c_ł).;                                                                                                                                                                      ].


Finally  we  apply Deift-Zhou steepest descend method   to get rid of the highly oscillatory terms in k in the  diagonal  exponents of the above matrix.
 For the purpose we open lenses in the intervals (ı c_ł,ı d) and (-ı d,-ı c_ł).
We first  need to define the analytic extension of the function f(k)=ı/a_+(k)a_-(k)  to a neighbourhood of the interval [ı c_ł,-ı c_ł].  Recalling that a_-(k)=-ıb(k) for k∈ (ı c_-,ı c_+)∪ (-ı c_+,-ı c_-) we define



    f(k) = -1/a(k)bk=1+r(k)rk/-rk,     k∈ U_σ([ı c_-,-ı c_-]).

Then  it is immediate to verify that 

    f_+(k)=f(k),  f_-(k)=-f(k),   k∈ [ı c_ł,ı c_]∪ [-ı c_,-ı c_ł].

 In order to get a factorization of the matrices J_X(x,t;  k) for k ∈ (ı c_ł,ı d)∪ (-ı d,-ı c_ł) we assume that
 
    T_+(k)T_-(k)=-ıf_+(k)=-ı f(k),   k∈ (ı c_ł,ı d),
        T_+(k)T_-(k)=-ıf( k),   k∈∪ (-ı d,-ı c_ł).


 In this way we can factorize
 
    J_X(ξ,t;k)=[                            1 T_+^2(k)^-2ı t g_+(k)/f_+(k);                            0                            1 ][ 0 ı; ı 0 ][                             1 -T_-^2(k)^-2ı t g_-(k)/f_-(k);                             0                             1 ],
       k∈ı c_ł,ı d,

    J_X(ξ,t;k) = [                          1                          0; -^2ı t g_+(k)/f_+kT_+^2(k)                          1 ][ 0 ı; ı 0 ][                         1                         0; ^2ı t g_-(k)/f_-kT_-^2(k)                         1 ],
       k∈(-ı d,-ı c_ł).

Then we proceed with contour deformation  and we define a new matrix W(k)  as

    W(k)=X(k)[                      1 T^2(k)^-2ı t g(k)/f(k);                      0                      1 ],   k∈Ω_5∪Ω_7,
    [                    1                    0; -^2ı t g(k)/fkT^2(k)                    1 ],   k∈Ω_6∪Ω_8,
    
    I

We obtain that the jumps for W(ξ,t;k) are

    J_W(ξ,t;k)=
    [                      1 T^2(k)^-2ı t g(k)/f(k);                      0                      1 ],   k∈ L_7,
        [                       1 T^2(k)^-2ı t g(k)/-f(k);                       0                       1 ],   k∈ L_5,
    [                       1                       0; -^2ı t g(k)/f(k)kT^2(k)                       1 ],   k∈ L_8,
        [                      1                      0; ^2ı t g(k)/f(k)kT^2(k)                      1 ],   k∈ L_6,
    [ 0 ı; ı 0 ]    k∈(ı c_ł,ı d)∪ (ı c_,-ı c_)∪(-ı d, -ı c_ł),
    
    e^ı (t B(ξ)+Δ(ξ))σ_3,   k∈(ı d,ı c_+)∪ (-ı c_,-ı d),

and for those values of   ξ   for which  -δ< g(κ_ℓ,ξ)<δ  for some point κ_ℓ of the discrete spectrum

    Res_κ_ℓW(ξ,t;k)=lim_k→κ_ℓW(ξ,t;k)[                        0                        0; ıν_ℓ/T^2(k)^2ı t  g(k,ξ)                        0 ],

    Res_-κ_ℓW(ξ,t;k)=lim_k→ -κ_ℓW(ξ,t;k)[                         0                         0; ı ν_ℓ/T^2(k)^2ı t  g(k,ξ)                         0 ],

    Res_κ_ℓW(ξ,t;k)=lim_k→κ_ℓW(ξ,t;k)[                         0 ı ν_ℓT^2(k)^-2ı t  g(k,ξ);                         0                         0 ],

    Res_-κ_ℓW(ξ,t;k)=lim_k→ -κ_ℓW(ξ,t;k)[                        0 ıν_ℓ T^2(k)^-2ı t g(k,ξ);                        0                        0 ].



Because of the signature of g(k) given in Figure <ref>, we have that the  matrix J_W in (<ref>) is exponentially close to the identity on L_5∪ L_7\{ U_ı d∪ U_ı c_ł} and  on L_6∪ L_8\{U_-ı d∪ U_-ı c_ł},
while on the  contours L_5∪ L_7∩{U_ı d∪ U_ı c_ł} and  on L_6∪ L_8∩{U_-ı d∪ U_-ı c_ł} the matrix J_W is close to the identity  but not uniformly. 
A detailed analysis of the error term arising in this case has been obtained in a similar setting for the MKdV equation with c_+=0<cit.> and also for the KdV equation in <cit.> where it was shown that the error is of order 𝒪(t^-1).
 


We arrive at the model problem for  the matrix P^∞(k). 

 Find a 2× 2 matrix-valued  function P^∞(ξ,t;k) analytic in k∈\ [ı c_-,-ı c_-] such that

  * P^∞_-(ξ,t;k)=P^∞_+(ξ,t;k)J_P^∞(ξ,t;k),   for k∈ (ı c_-,-ı c_-)
with 

    J_P^∞(ξ,t;k)={[ [ 0 ı; ı 0 ],
        k∈(ı c_-,ı d)∪ (ı c_+,-ı c_+)∪(-ı d, -ı c_-),;           ^ı( t B(ξ)+ Δ(ξ))σ_3,   k∈(ı d,ı c_+)∪ (-ı c_+,-ı d); ].

with B(ξ) and Δ(ξ) defined in (<ref>)   and (<ref>) respectively;

  *   for those values of   ξ   for which  -δ<Im  g(κ_ℓ,ξ)<δ, 0<δ≪1, 
 
    Res_κ_ℓP^∞(ξ,t;k)=lim_k→κ_ℓP^∞(ξ,t;k)[                        0                        0; ıν_ℓ/T^2(k)^2ı t  g(k,ξ)                        0 ],

    Res_-κ_ℓP^∞(ξ,t;k)=lim_k→ -κ_ℓP^∞(ξ,t;k)[                         0                         0; ı ν_ℓ/T^2(k)^2ı t  g(k,ξ)                         0 ],

    Res_κ_ℓP^∞(ξ,t;k)=lim_k→κ_ℓP^∞(ξ,t;k)[                         0 ı ν_ℓT^2(k)^-2ı t  g(k,ξ);                         0                         0 ],

    Res_-κ_ℓP^∞(ξ,t;k)=lim_k→ -κ_ℓP^∞(ξ,t;k)[                        0 ıν_ℓ T^2(k)^-2ı t g(k,ξ);                        0                        0 ];

  * P^∞(ξ,t;k)→ I   k→∞;

  * P^∞(ξ,t;k)  has at most fourth root singularities at the  points ±ı c_-, ±ı c_+ and ±ı d.


Then the quantity

    q(x,t)=lim_k→∞2ik P^∞_12(ξ,t;k)=lim_k→∞2ikP^∞_21(ξ,t;k),  ξ=x/12t,

approximates the MKdV solution for large times.   The   solution of the RH  problem <ref>,  excluding the pole condition 2,
  has been considered in <cit.> where it was solved in terms of hyperelliptic theta-functions defined on the Jacobi variety of the surface 
Γ:={(k,y)∈^2 |  y^2=R^2(k)}.  In Theorem <ref>  we showed that the MKdV solution  obtained from   the RH problem <ref> without the pole condition 2  corresponds to the travelling wave solution 
(<ref>). In particular we obtain  that for V_ℓ+δ̃<ξ< c_ł^2/3+c_^2/6 and -c_ł^2/2+c_^2<ξ<V_ℓ-δ̃ for some δ̃>0,   namely for ξ distinct from the 
breather speed V_ℓ we have 

    q(x,t)=q_per(x,t;c_-,d,c_+,x_0)+O(t^-1),

where   q_per(x,t;β_1,β_2,β_3,x_0)  is the travelling wave defined in (<ref>) and 

    x_0=-K(m)Δπ+K(m)

with K(m) the complete elliptic integral   with modulus m^2=d^2-c_+^2c_-^2-c_+^2 and Δ defined in (<ref>).
The speed  V_ℓ  of the breather is obtained by solving the equation g(κ_ℓ,V_ℓ)=0 and  by (<ref>) and (<ref>) is 

    V_ℓ=-∫_-c_-^2^κ_ℓ^2s^2+s/2(c_-^2+c_+^2+d^2)+b_1/√(s+c_-^2)(s+c_+^2)(s+d^2)) ds∫_-c_-^2^κ_ℓ^2s+b_0/√(s+c_-^2)(s+c_+^2)(s+d^2)) ds,

where b_0=c_ł^2-(c_ł^2-c_^2)E(m)K(m) and 

    b_1=13(c_ł^2c_^2+c_ł^2d^2+c_^2d^2)-1/6(c_ł^2+c_^2+d^2)[c_ł^2-(c_ł^2-c_^2)E(m)K(m)].


 When  |V_ℓ-ξ|<δ̃  the solution  of the RH problem <ref>  with the pole condition 2 describes a breather on the elliptic background that we indicate as q_be(x,t).
The determination of the explicit expression of the breather on the elliptic background  is  similar to the derivation obtained in Section <ref> for the breather on the constant background, but the computation is  algebraically involved and it is omitted.

The error term in (<ref>) is  similar  to the  case  computed in <cit.> where all the local parametrices are the same. We add the derivation in the section below.
The error term is valid strictly inside the dispersive shock wave region. On the boundary 
of the dispersive shock wave region a different error term is present (see <cit.>) for details).
We have thus concluded the proof of  the Theorem <ref>  part (b)  leading order term.



  §.§.§ Estimate of the error term  in the dispersive shock wave region.

This section is fully analogous to Section 3 in <cit.>.
We calculate the error term in the dispersive shock wave region away from the breathers.
The jump matrix J_W has the main jump over the segment [ı c_-,-ı c_-], and on the rest of the contour is exponentially close to the identity matrix except for small neighbourhoods of the points ±ı c_- and ±ı d. This means that we need to construct local solutions to the RH problem for the function W near the points ±ı c_-, ±ı d.

To deal with the point ı c_-, we `push up' the contours L_7 and L_5, so that they join at a point ı (c_-+ ε) (with some small ε>0) instead of the point ı c_-. In this way we completely remove the problem at the point ı c_- (this corresponds to the trivial case of the `Bessel' parametrix.)

The point ı d requires more careful considerations. First, we deform  the contour L_1  downward, so that it  intersects the segment [ı c_-,-ı c_-] at a point ı d-ıε, ε>0 instead of the point ı d. 
Then, we construct an explicit exact solution of the W-RH problem inside the disk |k-ı d|<ε in terms of the Airy parametrix. This will give us an error term of the order t^-1. 
Below are the details.
Point k=ı d(ξ). Due to the symmetry k→ k, it is enough to consider the point ı d(ξ) only, and then the construction near the point -ı d(ξ) will follow by symmetry.

Note that the phase function g(k,ξ)=∫_ı c_-^k g'(s,ξ)ds   with g'(k,ξ) as in (<ref>).
Since g'(k,ξ) vanishes as √(k-ı d) when  k→ı d it follows that near the point k=ı d    we have  g(k,ξ)- ∫_ı c_-^ı d g'(s,ξ)ds∼(k-ı d)^3/2.

For δ>0 we define the neighbourhood

    𝒰_ı d={k∈, s.t.  |k-ı d|<δ}

and 𝒰_ı d^± are the right and left neighbourhood of ı d with respect to the imaginary axis.
Next we define the conformal map

    ζ=(-3/2ı [tg(k)- t∫_ı c_-^ı d g'_±(s,ξ)ds])^2/3,    s,k∈𝒰_ı d^±.









































Furthermore, we introduce the function

    ϕ(k, ξ)=T(k, ξ)^ı t B(ξ)/2/√(-ıf(k)),    k∈𝒰_ı d^+,
    
    
    T(k, ξ)^-ı t B(ξ)/2/√(ıf(k)),    k∈𝒰_ı d^-,

which has a jump across k∈(ı (d+δ),-ı (d-δ)).   We recall that f(k) is defined in (<ref>), B(ξ) is defined in (<ref>)  and T(k,ξ) is defined in (<ref>). We also observe   that -ıf_+(k)>0 and ıf_-(k)>0 for k∈(ı c_-, ı c_+).
Then the jump matrix J_W(k)=J_W(ξ,t;k) in (<ref>) in 𝒰_ı d  takes the following form:

    J_W(k)=ϕ^σ_3[           1 -ı^2/3ζ^3/2;           0           1 ]ϕ^-σ_3,   k∈ (L_7∪ L_5)∩𝒰_ı d,
    
       
    J_W(k)=ϕ_+^σ_3[           1           0; ı^-2/3ζ^3/2           1 ]ϕ_-^-σ_3,   k∈ (ı d, ı (d-δ)),
       
    J_W(k)=ϕ_+^σ_3[ 0 ı; ı 0 ]ϕ_-^-σ_3,   k∈(ı (d+δ),ı d).
Local parametrix at ı d(ξ).  In order to mimic the above jumps matrix, we introduce the matrix function P_Ai(k) obtained via the Airy function.

    P_Ai(k):=Ψ_Ai(ζ)ϕ(k,ξ)^-σ_3^-2/3ζ^3/2σ_3,

where Ψ_Ai(ζ) is defined as follows:


    Ψ_(ζ)
    =[  v_1(ζ)  v_0(ζ); v'_1(ζ) v'_0(ζ) ],ζ∈(0,2/3π),
    
    [      v_1(ζ)  -ı v_-1(ζ);     v'_1(ζ) -ı v'_-1(ζ) ],ζ∈(2/3π,π),
         
    [   v_-1(ζ)  ı v_1(ζ);  v'_-1(ζ) ı v'_1(ζ) ],ζ∈(-π, -2/3π),
    
    [  v_-1(ζ)   v_0(ζ); v'_-1(ζ)  v'_0(ζ) ],ζ∈(-2/3π, 0),

where the functions
v_0, v_1, v_-1 are defined in terms of the Airy function  as 

    v_0(ζ)=√(2π)(ζ),   
    v_1(ζ)=^-πı/6√(2π)(ζ^-2πı/3),
      
    v_-1(ζ)=^πı/6√(2π)(ζ^2πı/3),

and ' denotes the derivative in ζ. The well-known relation (ζ)+^2πı/3(ζ^2πı/3)+^-2πı/3(ζ^-2πı/3)=0 reads as

    v_0(ζ)-ı v_1(ζ) + ı v_-1(ζ)=0.

The latter allows us to verify that the function Ψ_ has the following jumps Ψ_,-(ζ)=Ψ_,+(ζ)J_(ζ) across the contour Σ_=ℝ∪(^2πı/3∞, 0)∪(^-2πı/3∞, 0), which is oriented according to the order in which the points are mentioned, that is from ^±2πı/3∞ to 0 and from -∞ to +∞,
    J_(ζ) = [ 1 0; ı 1 ], ζ=0,  
    J_(ζ) = [  1 -ı;  0  1 ], ζ=±2π/3,  
       J_(ζ) = [ 0 ı; ı 0 ],ζ=π.

Besides, the function Ψ_ admits the following asymptotics for large ζ, which is uniform with respect to ζ∈[-π,π]:

    Ψ_(ζ)=1/√(2)ζ^-σ_3/4[  1  1;  1 -1 ]I+[ -1/48  -1/8;   1/8  1/48 ]ζ^-3/2 + 𝒪(ζ^-3)^2/3ζ^3/2σ_3,  ζ→∞.
Approximation of W.
Now we define a function W_appr(ξ,t;k), which will be shown to be a good approximation of the function W(ξ,t;k),
    W_appr(ξ,t;k)
    =
    P^(∞)(k),   |k∓ı d(ξ)|>δ,
    
    ℬ(k) Ψ_(ζ(k))ϕ(k,ξ)^-σ_3^-2/3(ζ(k))^3/2σ_3,
       |k-ı d(ξ)|<δ,
    
    
    W_appr(ξ, t;  k),   |k+ı d(ξ)|<δ.

Note that here we defined W_appr for |k+ı d|<δ in terms of W_appr for |k-ı d|<δ.
Furthermore, here ℬ is an unknown matrix  function  analytic inside the disk |k-ı d|<δ. It is obtained by requesting that 
the error 
    E(k) = W(k) W_appr(k)^-1
 has jump J_E(k)=I+o(1) as t→∞ and k∈∂𝒰_ı d.
Hence, using (<ref>), we obtain  

    ℬ(k)
    =
    P^(∞)(k)ϕ(k,ξ)^σ_31/√(2)[  1  1;  1 -1 ]ζ^σ_3/4.

Note that ℬ(k) is indeed holomorphic in the disk 𝒰_ı d. This can be verified using the relations ϕ_+ϕ_-=1 for k∈(ı (d+δ), ı d), and ϕ_+=ϕ_-^ı t B+ıΔ for k∈(ı d, ı (d-δ)).

Using the asymptotics   (<ref>), we see that on the circle  ∂𝒰_ı d the jump matrix for the error function E_-(k)=E_+(k)J_E(k)   is given by

    J_E(k)=
    P^(∞)(k)ϕ(k,ξ)^σ_31/√(2)[  1  1;  1 -1 ]ζ^σ_3/4Ψ_(ζ)
    ϕ(k)^-σ_3^-2/3ζ^3/2σ_3
    P^(∞)(k)^-1,

is indeed of the order I+𝒪(ζ^-3/2)=I+𝒪(t^-1)  as t→∞, uniformly on the contour.
 Hence the error matrix E(k)  is also of the same order, namely E(k)=I+𝒪(t^-1).
 Therefore for large |k| we have W(k)= P^(∞)(k)+W_err(k) where W_err(k) is of the order 𝒪(t^-1).
The solution q(x,t) of the MKdV equation equals

    q(x,t)=lim_k→∞(2ı kW(k))_21
    =
    lim_k→∞(2ı kP^(∞)(k))_21+
    lim_k→∞(2ı kW_err(k))_21,

where, in the r.h.s.,  the second term is of the order 𝒪(t^-1), while the first term is q_ell(x,t).
We thus obtain

    q(x,t)=q_ell(x,t)+𝒪(t^-1)

for ξ≡x/12t strictly inside the elliptic region, -c_-^2/2+c_+^2+δ≤ξ<c_-^2/3+c_+^2/6-δ.

§ LARGE TIME ASYMPTOTICS: PROOF OF THEOREM <REF>,  PARTS (A) AND (C)


Here we continue the proof of Theorem <ref> in the soliton-breather region and in the breather region.










 §.§ Breather region: Theorem <ref>  part (c)

In the breather region the function  g(k,ξ) takes the form  <cit.>
    g(k,ξ)=22k^2-c_-^2+6ξ)√(k^2+c_-^2) ,    ξ<-c_-^2/2+c_+^2,
    
    namely g(k,ξ) is analytic in ℂ\ [ı c_-,-ı c_-]  and
    
        g_+(k,ξ)+g_-(k,ξ)=0,   k∈(ı c_-,-ı c_-),   g(k,ξ)=θ̂(k,ξ)+O(k^-1)  .
    
    The above two conditions defined g(k,ξ) uniquely.  We chose √(k^2+c_-^2) to be real on (ı c_-,-ı c_-) and positive for k=+0.
    
    We observe that g(k,ξ)=0 for k on the segment [ı c_-,-ı c_-] and on the real line. 
    For a given ξ, there is an extra   couple of  symmetric curves such that   g(k,ξ)=0   (see  Figure <ref>).  For  ξ<-c_-^2/2 this   couple of  curves  
     crosses the real  line  at the points k=± k_0=±√(-c_-^2/2-ξ)   (utmost left constant region). For   -c_-^2/2<ξ<-c_-^2/2+c_+^2  the level lines
     g(k,ξ)=0 
    are  a couple of  symmetric curves that  cross  the imaginary axis at the points k=±ı d_0=±ı√(ξ+c_-^2/2)   (middle left  constant region).
    The points ± k_0 and ±ı d_0  are the zeros of   the equation
    
        g'(k,ξ)=6(2k^2+c_-^2+2ξ)k/√(k^2+c_-^2)=0 .
    
    
    
    The asymptotic analysis in these two cases is slightly different. We start our analysis  with the middle left constant region.
    
    
      §.§.§ Middle left constant region -c_ł^2/2<x/12t≡ξ<-c_ł^2/2+c_^2
    
    
    We define  the first transformation of the RH problem  <ref>, M(k)→ Y(k),  defined as 
    
    
        Y (k)=M(k)^ıtg(k,ξ)-θ(x,t;k)σ_3T^-σ_3(k,ξ),
    
    where now g(k,ξ) is as in (<ref>)  and T(k)=T(k,ξ) is defined  for a given ξ as follows:
    
        T(k,ξ)   =T(k,ξ)H(k,ξ),
        T(k,ξ)    = ∏_[ κ_j>0,κ_j>0;  g(κ_j,ξ)<0 ]k-κ_j/k-κ_jk+κ_j/k+κ_j·∏_[ κ_j=0,κ_j>0;  g(κ_j,ξ)<0 ]k-κ_j/k-κ_j,
    
    and H(k,ξ)  is supposed to be analytic in ℂ\ [ı c_-,-ı c_-] and it will be determined later. 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    In order to perform our analysis on the continuum spectrum we assume
     the condition (<ref>)  for g(k) and 
    
        T_+(k)T_-(k)=
        1/|a(k)|^2,   k∈(ı c_ł, ı d_0), 
         |a(k)|^2,   k∈(-ı d_0,-ı c_ł), 
         1,   k∈(ı d_0,-ı d_0) .
    
    Then the  function T(k) can be found  taking the logarithm  of the above expression  and applying Plemelj formula so that we obtain
    
    
    
    
    
    
    
    
    
    
    
    
    
        T(k,ξ)=T(k,ξ) 
        exp   [√(k^2+c_ł^2)/2πı{∫^-ı c_ł_-ı d_0-∫_ı c_ł^ı d_0ln|a(s)|^2ṣ/(s-k)(√(s^2+c_ł^2))_+-..
           ..-
        ∫_ı c_-^-ı c_-lnT^2(s,ξ)ṣ/(s-k)(√(s^2+c_ł^2))_+}] .
    
    
    
     Let us notice, that T(k,ξ)=1+𝒪1/k, as k→∞.
    
    
    With our choice of g(k) and T(k)  as in (<ref>) and (<ref>),  respectively,  we consider the matrix Y(k) defined in (<ref>). Then it is clear that the jumps  J_Ỹ  as in 
    (<ref>) on the circles 
    C_j, C_j, -C_j and -C_j are exponentially small
     for those values of κ_j for which  g(κ_j,ξ)>δ or  g(κ_j)<-δ, κ_j≥ 0 and κ_j>0.
         The values of κ_j   and ξ  for which -δ< g(κ_j,ξ)<δ  will be considered later.  Now we take care of the continuous spectrum.
          The jump matrix J_Ỹ(ξ,t;k)   relative to the RH problem for Y (ξ,t;k), Y_-(ξ,t;k)=Ỹ_+(ξ,t;k)J_Ỹ(ξ,t;k)  admits, for k≠±ı c_+,  a  factorization   of the form 
    
        J_Ỹ(ξ,t;k)=
        [                            1 T_+^2(k)^-2ı t g_+(k)/f_+(k);                            0                            1 ][ 0 ı; ı 0 ][                             1 -T_-^2(k)^-2ı t g_-(k)/f_-(k);                             0                             1 ], k∈(ı c_ł,ı d_0)
        
        [                          1                          0; -^2ı t g_+(k)/f_+kT_+^2(k)                          1 ][ 0 ı; ı 0 ][                         1                         0; ^2ı t g_-(k)/f_-kT_-^2(k)                         1 ],
           k∈(-ı d_0,-ı c_ł)
        
        [                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ][                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ],   k∈ℝ\{0}
        [  ı r_-(k)T_+(k)^ı t (g_–g_+)/T_-(k)      ı  T_+(k)T_-(k)^-ı t (g_-+g_+);         f^ı t(g_-+g_+)/T_+(k)T_-(k) -ı r_+(k)T_-(k)^ı t(g_+-g_-)/T_+(k) ],   k∈ (ı d_0,0)
        
        [   ır_+kT_+(k)^ı t (g_–g_+)/T_-(k)          -fk T_+T_-^-ı t(g_-+g_+);      ı^ı t (g_-+g_+)/T_+(k)T_-(k) -ı r_-kT_-(k)^ı t(g_+-g_-)/T_+(k) ], 
         k∈(0,-i d_0)
    
    
    Here 
        f(k) = -1/a(k)bk=1+r(k)rk/-rk.
    
    
    
    Now we open the lenses as in Figure <ref> where we assume that we can deform all the curves L_j so that there are no poles within the regions
    Ω_j.
    We define a new matrix X as
    
        X(ξ,t;k)=Ỹ(ξ,t;k)[                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ],   k∈Ω_1,
        Ỹ(ξ,t;k)[                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ]^-1,   k∈Ω_2,
        Ỹ(ξ,t;k) [                          1                          0; -^2ı t g_+(k)/f_+kT_+^2(k)                          1 ],   k∈Ω_8,
        Ỹ(ξ,t;k)[                         1                         0; ^2ı t g_-(k)/f_-kT_-^2(k)                         1 ]^-1,   k∈Ω_6,
        Ỹ(ξ,t;k) [                            1 T_+^2(k)^-2ı t g_+(k)/f_+(k);                            0                            1 ],   k∈Ω_7,
        Ỹ(ξ,t;k)
        [                             1 -T_-^2(k)^-2ı t g_-(k)/f_-(k);                             0                             1 ]^-1,   k∈Ω_5.
    
    With the above transformation the jump matrix on (ı d_0, 0) transforms to 
    
        [                  1                  0; r_+^2ı t g_+/T_+^2                  1 ][     ı r_-T_+^ı t (g_–g_+)/T_-      ı  T_+T_-^-ı t (g_-+g_+);         f^ı t(g_-+g_+)/T_+T_- -ı r_+T_-(k)^ı t(g_+-g_-)/T_+ ][                   1                   0; -r_-^2ı t g_-/T_-^2                   1 ]
        =
        
           
        =[ 0 ı; ı 0 ], k∈(ı d_0,0),
    
    and  similarly on (0,-ı d_0).
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    We obtain that the matrix X(k) solves the following RH problem.
    Final RH problem for the middle left constant region -c_ł^2/2<ξ< -c_ł^2/2+ c_^2.
      *  Find a 2× 2 matrix X(k) meromorphic in ℂ\Σ with 
    Σ= {∪_j=1^7 L_j∪(ı c_-,-ı c_-)}  (see Figure <ref>)   such that 
    
      *  Jumps:
    
        X_-(ξ,t;k)=X_+(ξ,t;k)J_X(ξ,t;k),   k∈Σ
    
    and 
    
    
        J_X(ξ,t;k)=[ 0 ı; ı 0 ],   k∈(ı c_ł, -ı c_ł),
    
        J_X(ξ,t;k)=[                       1                       0; -r(k) ^2ı t g(k)/T(k)^2                       1 ], k∈ L_1,
        [                      1 -rk T(k)^2 ^-2ı t g(k);                      0                      1 ], k∈ L_2, 
        [                      1 T(k)^2^-2ı t g(k)/f(k);                      0                      1 ], k∈ L_7,  [                    1 -T^2(k)^-2ı t g/f(k);                    0                    1 ], k∈ L_5,
        [                     1                     0; -^2ı t g(k)/fk T(k)^2                     1 ], k∈ L_8,
          [                    1                    0; ^2ı t g(k)/fk T(k)^2                    1 ], k∈ L_6.
    
      *  Poles: for those κ_j with -δ< g(κ_j,ξ)<δ, κ_j>0, κ_j≥ 0:
        Res_κ_jX(ξ,t;k)=lim_k→κ_jX(ξ,t;k)[                        0                        0; ıν_j/T^2(k)^2ı t  g(k,ξ)                        0 ],
    
        Res_-κ_jX(ξ,t;k)=lim_k→ -κ_jX(ξ,t;k)[                          0                          0; ı ν_j /T^2(k)^2ı t  g(k,ξ)                          0 ],
    
        Res_κ_jX(ξ,t;k)=lim_k→κ_jX(ξ,t;k)[                         0 ı ν_jT^2(k)^-2ı t  g(k,ξ);                         0                         0 ],
    
        Res_-κ_jX(ξ,t;k)=lim_k→ -κ_jX(ξ,t;k)[                        0 ıν_j T^2(k)^-2ı t g(k,ξ);                        0                        0 ].
    
      *  Asymptotics: X(ξ,t;k)→ I as k→∞.
    
      §.§.§ Utmost left constant region ξ<-c_ł^2/2
    
    As in the other case we define
    
        Y (ξ,t;k)=M(x,t;k)^ıtg(k,ξ)-θ(x,t;k)σ_3T^-σ_3(k,ξ),
          ξ=x/12t,
    
    where the fuction  g(k) is as  in  (<ref>)  while for the function  T(k)   defined in (<ref>)  we require  the following conditions that are needed in order to 
    apply the Deift-Zhou steepest descent method to deform the contours:
    
        T_+(k)T_-(k)=-ı f(k)=1/|a(k)|^2, k∈(ı c_ł, 0), 
        -ı/f(k)=|a(k)|^2, k∈(0, -ı c_ł),
    
    and 
    
        T_+(k)/T_-(k)=1+|r(k)|^2,   k∈(-k_0, k_0),   k_0=√(-ξ-c_-^2/2),
    
    with  T(k)→ 1 as |k|→∞.  Observe that in this case T(k) has a cut across [ı c_-,-ı c_-] and also on a finite interval [-k_0,k_0] of the real axis.
    
     This function can be found in the form 
    
        T(k,ξ)=T(k,ξ) 
        exp[√(k^2+c_ł^2)/2πı∫_ı c_ł^0-ln|a(s)|^2-lnT(s,ξ)^2ṣ/(s-k)( √(s^2+c_ł^2))_+]·
           ·exp[√(k^2+c_ł^2)/2πı{-1mm∫_0^-ı c_łln|a(s)|^2-lnT(s,ξ)^2ṣ/(s-k)(√(s^2+c_ł^2))_+
        +
        -1mm
        ∫_-k_0^k_0ln1+|r(s)|^2ṣ/(s-k)(√(s^2+c_ł^2))_+}]
    
    with  T(k,ξ) as  in formula (<ref>).
    
     
    Because of (<ref>) and (<ref>), the jump matrix J_Y of the RH problem for Y(k)
    has a factorization on the real axis of the form
    
        J_Y(ξ,t;k)=
        [                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ][                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ],  k∈ℝ\(-k_0, k_0)
        [                         1 -rk T_+^2^-2ı t g/1+|r|^2;                         0                         1 ][                            1                            0; -r(k) ^2ı t g/(1+|r|^2)T_-^2                            1 ],    k∈(-k_0, k_0) .
    
    We use the above factorization  to  open the lenses and  define the matrix X(ξ,t;k) as
    
        X(ξ,t;k)=
           Y(ξ,t;k)[                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ],   k∈Ω_1, 
           Y(ξ,t;k)[                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ]^-1,   k∈Ω_2, 
           Y(ξ,t;k)[                         1 -rk T_+^2^-2ı t g/1+|r|^2;                         0                         1 ],   k∈Ω_3, 
           Y(ξ,t;k)[                               1                               0; -r(k) ^2ı t g/(1+|r|^2)T_-^2(k)                               1 ]^-1,    k∈Ω_4, 
           Y(ξ,t;k),   .
    
    We arrive at  the following RH problem for the function X(ξ,t;k).
    Final RH problem for the utmost left  constant region ξ<-c_ł^2/2.
      *  Find a 2× 2 matrix function X(ξ,t;k) meromorphic  for k∈ℂ\Σ with Σ=∪_j=1^4L_j∪(ı c_-,-ı c_-)  (see Figure <ref>) 
    and   such that
    
      * X_-(ξ,t;k)=X_+(ξ,t;k)J_X(ξ,t;k),   k∈Σ
    with 
    
        J_X(ξ,t;k)=[ 0 ı; ı 0 ], k∈(ı c_ł, -ı c_ł),
    
        J_X(ξ,t;k)=[                       1                       0; -r(k) ^2ı t g(k)/T^2(k)                       1 ], k∈ L_1,
        [                      1 -rk T^2(k) ^-2ı t g(k);                      0                      1 ], k∈ L_2, 
        [                              1 -rk T^2(k)^-2ı t g(k)/1+r(k)rk;                              0                              1 ], k∈ L_3,
        [                               1                               0; -r(k) ^2ı t g(k)/1+r(k)rkT^2(k)                               1 ], k∈ L_4,
    
    and the pole conditions are the same as in the RH problem <ref>.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
      *  Asymptotics: X(k)→ I as k→∞.
    
      §.§.§ Model problems for the regions ξ<-c_ł^2/2 and -c_ł^2/2<ξ<-c_ł^2/2+c_^2
    
    Looking at the RH problems <ref> and  <ref>, we observe that the jump matrix is exponentially close to I everywhere, except for the jump on (ı c_ł, -ı c_ł), and the parts of the curves L_j,j=1,…,8, intersecting either the real line (i.e. the vicinities of the points ± k_0(ξ)) , or the interval (ı c_ł, -ı c_ł) (i.e. the vicinities of the points ±ı d_0(ξ)). A careful analysis of the contribution of the points ±k_0(ξ), to the asymptotics for q(x,t) involves the construction  of a local (approximate) solution to the RH problem in the vicinities of those points, and is usually called  parametrix analysis. The parametrices in the vicinities of the points ± k_0 can be constructed in terms of parabolic cylinder function, in a similar way as  in <cit.>, and they give the contribution of the order 𝒪(t^-1/2). The parametrices in the vicinities of the points ±ı d_0 can be constructed in terms of the Airy function and its derivative, in a similar way  as was done in <cit.>, and they give a contribution of the order 𝒪(t^-1). We thus reduce the problem to analysis of the corresponding model problems.
    
    The regular RH model problem in these regions gives the constant c_ł as the main term in the asymptotic expansion  for q(x,t).
    The meromorphic RH model  problems give us the breathers  on a constant background.
    
    The model problem for this case is 
    Model RH problem for the 
    
    middle and utmost left
    constant regions -c_ł^2/2<ξ<-c_ł^2/2+c_^2   and -c_ł^2/2<ξ. 
    
    Find a 2× 2 matrix M^mod(k,ξ) meromorphic for  k∈ℂ\[ı c_-,-ı c_-] and such that
    
      *  Jump: 
        M^mod_-(k)=M^mod_+(k)[ 0 ı; ı 0 ],   k∈(ı c_ł, -ı c_ł),
    
      *  Pole: if there are points κ_j with κ_j>0,κ_j>0,  -δ< g(κ_j,ξ)<δ, we have the pole condition
    
        Res_κ_jM^(mod)(k)=lim_k→κ_jM^(mod)(k)[                          0                          0; ıν_j/T^2(k,ξ)^2ı t  g(k,ξ)                          0 ],
    
        Res_-κ_jM^(mod)(k)=lim_k→ -κ_jM^(mod)(k)[                            0                            0; ı ν_j /T^2(k,ξ)^2ı t  g(k,ξ)                            0 ],
    
        Res_κ_jM^(mod)(k)=lim_k→κ_jM^(mod)(k)[                           0 ı ν_jT^2(k,ξ)^-2ı t  g(k,ξ);                           0                           0 ],
    
        Res_-κ_jM^(mod)(k)=lim_k→ -κ_jM^(mod)(k)[                          0 ıν_j T^2(k,ξ)^-2ı t g(k,ξ);                          0                          0 ],
    
      * M^(mod)(k)=1+O(k^-1) as |k|→∞.
    
    We observe that  g(κ_j,ξ)=( √(κ_j^2+c_-^2))(12ξ-V_j) where 
        V_j=4( √(κ_j^2+c_-^2))^2+6c_-^2-12( √(κ_j^2+c_-^2))^2
       is the speed of the breather corresponding to the spectrum κ_j on the constant background c_-.
    
    Therefore the condition  -δ< g(κ_j,ξ)<δ,  is equivalent to requiring that 
    
        |V_j-12ξ|<δ̃,  ξ=x12 t,   δ̃=δ/( √(κ_j^2+c_-^2)).
     
    Namely we can see the  breather if  we observe in the direction of the (x,t)   plane such that |V_j-x/t|<δ̃.   
    We number the  points of the discrete spectrum κ_j, according to their speed as specified in  the introduction.
    
    
    
    
    
    
    
    We further observe that if ξ is such that  g(κ_j,ξ)=0 then   ξ=V_j/12 and  with the above ordering we have that 
    
        g(κ_l,V_j/12)<0,  g(κ_l,V_j/12)>0,   .
    
    With this ordering the function T(k,V_j/12) defined in (<ref>) takes the form (<ref>).
    The RH problem  <ref>  corresponds to the case of a single breather on a constant background  that   we considered in Section <ref>.
     This problem can be solved explicitly in terms of elementary functions, and 
    
        2ılim_k→∞M_12^(mod)(k,ξ) = q_breath(x,t; c_ł,κ_j, ν_j) ,
    
    where
    
        ν_j = ν_j T^2(κ_j,ξ),
    
     with T(k,ξ) as in (<ref>) 
       for -c_ł^2/2<ξ<-c_ł^2/2+c_^2     and d_0=ı√(ξ+c_-^2/2), and T(k,ξ) as in (<ref>)  for 
     ξ>-c_ł^2/2, k_0=√(-ξ-c_-^2/2).
    Summing up, we have shown that for  large values of time the solution of the MKdV equation 
    in the domains ξ<-c_ł^2/2 and -c_ł^2/2<ξ<-c_ł^2/2+c_^2   is as follows:
    
      (1) for those ξ, t such that  | g(κ_j,ξ)|>δ for all κ_j we have
    
        q(x,t)=c_ł+𝒪(t^-1/2),
    
      (2) For those ξ, t such that there exists κ_j, κ_j>0,κ_j>0,  with | g(κ_j, ξ)|<δ,
    we have 
    
        q(x,t)=q_breath(x,t; c_ł, κ_j, ν_j)+𝒪(t^-1/2),
    
    where q_breath has been defined in (<ref>) and the phase   ν_j in (<ref>).
    
    Thus we have concluded the proof of  Theorem <ref>   part (c).
    
    
    
    
    
    
     §.§ Proof of Theorem <ref>   part (a):   soliton and breather region
    
    In this case we consider the right constant region ξ>c_ł^2/3+c_^2/6  where solitons and breathers are moving in the positive x direction with a speed greater then the dispersive shock wave.
    As in the  previous case we introduce the matrix function
    
        Y (ξ,t;k)=M(x,t;k)^ıtg(k,ξ)-θ(x,t;k)σ_3T^-σ_3(k,ξ),
          ξ=x/12t,
    
    where now the function  g(k,ξ) takes the form  <cit.>
        g(k,ξ)=22k^2-c_+^2+6ξ)√(k^2+c_+^2) ,     
        namely g(k,ξ) is analytic in k∈ℂ\ [ı c_+,-ı c_+]  and
        
            g_+(k,ξ)+g_-(k,ξ)=0,  k∈(ı c_+,-ı c_+),   g(k,ξ)=4k^3+12ξ k
            +O(k^-1), .
        
         We chose √(k^2+c_+^2) to be real on (ı c_+,-ı c_+) and positive for k=+0.
        The sign of g(k,ξ) are plotted in Figure <ref>.
        
        The function T(k,ξ) is as in (<ref>)  with c_- replaced by c_+ and it satisfies the condition 
        
            T_-(k)T_+(k)=1,   k∈(ı c_ł, -ı c_ł) .
        
        The solution is  explicitly given as follows:
        
            T(k,ξ)=T(k,ξ)
             exp[√(k^2+c_^2)/2πı∫_ı c_^-ı c_-lnT^2(s,ξ)ṣ/(s-k)( √(k^2+c_^2))_+],
        
        where T(k,ξ) is of the form (<ref>). 
        This choice of the function T(k,ξ) permits the factorization of the jump matrix J_Y on the real axis in the form 
        
            J_Y(ξ,t;k)=[                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ][                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ],   k∈ℝ.
        
        We use the above factorization  to  open the lenses and  define the matrix X(ξ,t;k) as
        
            X(ξ,t;k)=
               Y(ξ,t;k)[                        1                        0; -r(k)^2ı t g(k,ξ)/T^2(k)                        1 ],   k∈Ω_1, 
               Y(ξ,t;k)[                      1 -rkT^2(k)^-2ı t g(k,ξ);                      0                      1 ]^-1,   k∈Ω_2, 
               Y(ξ,t;k),   ,
        
        where the regions Ω_1 and Ω_2 are specified Figure <ref>.
        
        Final RH problem for the soliton and breather region ξ>c_ł^2/3+c_ł^2/6.
        Find a 2× 2 matrix X(ξ,t;k) meromorphic in k∈ℂ\Σ where the contour Σ is specified below
        and 
        
            X_-(ξ,t;k)=X_+(ξ,t;k)J_X(ξ,t;k),   k∈Σ,
        
        and 
        
            J_X(ξ,t;k)=
            [ 0 ı; ı 0 ], k∈(ı c_, -ı c_),
            
             J^(2)=I,  k∈(ı c_ł, ı c_) ∪ (-ı c_, -ı c_ł),
            [                           1                           0; -r(k) ^2ı t g(k,ξ)/T^2(k,ξ)                           1 ],    k∈ L_1,
            [                          1 -rk T^2(k,ξ) ^-2ı t g(k,ξ);                          0                          1 ],    k∈ L_2,
        
        with L_1 and L_2 as in figure <ref>  and 
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        at the points κ_j with κ_j>0,  we keep the pole condition  as in the RH problem <ref>.
        
        
        
        
        
        
        
        
        Finally the normalizing condition at infinity is  X(k)=1+O(k^-1) as |k|→∞.
        
        We observe that the jump matrices  approach the identity exponentially fast on L_1 and L_2  and also for those values of ξ and κ_j such that κ_j>0  and 
        g(κ_j)>ε or g(κ_j)<-ε.
         We finally arrive to the following model problem.
        Model RH problem for the right region  ξ>c_ł^2/3+c_^2/6.
        Find a 2× 2 matrix M^mod(ξ,t;k) meromorphic for  k∈ℂ\[ı c_+,-ı c_+] and such that
        
          *  Jump: 
            M^(mod)_-(ξ,t;k)=M^(mod)_+(ξ,t;k)[ 0 ı; ı 0 ], k∈(ı c_, -ı c_).
        
          *  Poles: if there are points κ_j with κ_j>0,κ_j>0,  -δ< g(κ_j,ξ)<δ, we have the pole condition
        
            Res_κ_jM^(mod)(ξ,t;k)=lim_k→κ_jM^(mod)(ξ,t;k)[                        0                        0; ıν_j/T^2(k)^2ı t  g(k,ξ)                        0 ],
        
            Res_κ_jM^(mod)(ξ,t;k)=lim_k→κ_jM^(mod)(ξ,t;k)[                         0 ı ν_jT^2(k)^-2ı t  g(k,ξ);                         0                         0 ],
        
        and the  symmetric conditions at the points -κ_j and  -κ  when we are considering a breather.
        
          * M^(mod)(ξ,t;k)=1+O(k^-1) as k→∞.
        
        
        This model problem is exactly the model problem, which gives solitons or breathers on the constant background c_.
        Thus,
        
            lim_k→∞2ı k(M^mod(ξ,t;k))_12 = q_sol(x,t; c_, κ_j, x_j),  κ_j=0,
            
               lim_k→∞2ı k(M^mod(ξ,t;k))_12 = q_breath(x,t; c_, κ_j, ν̂_j),  κ_j>0,
        
        where ξ=x/(12t) and where  we use the  numbering of solitons and breathers  according to their velocities  so that the phase shift can be written in the form
        
            ν̂_j = ν_j T_j^2(κ_j),   x_j=log2(κ_j^2-c_+^2)T_j^2(ı|κ_j|)|ν_j|κ_j, 
            
               T_j(k):=T(k,V_j/12)=T(k,V_j/12)
             exp[√(k^2+c_^2)/2πı∫_ı c_^-ı c_-lnT^2(s,ξ)ṣ/(s-k)( √(k^2+c_^2))_+],
        
        where   T(k,ξ)  as in (<ref>)  so that T(k,V_j/12) coincides with T_j(k) defined in (<ref>).
        
        
        
        
        
        
        Since the jump matrices are exponentially close to the identity matrix uniformly, the contribution of the poles give the leading order asymptotic expansion, and the whole contours give exponentially small contribution. We can conclude that the solution of the Cauchy problem  (<ref>), (<ref>) in the domain ξ>c_ł^2/3 + c_^2/6
        has the following asympotics as t→∞:
          (1) For those ξ, t such that  | g(κ_j,ξ)|>δ for all κ_j we have
        
            q(x,t)=c_+𝒪(^-C t),
        
        for some constant C>0.
        
          (2) For those ξ, t such that there exists κ_j,  κ_j>0, with | g(κ_j, ξ)|<δ, 
        we have 
        
            q(x,t)=
               q_breath(x,t; c_ł, κ_j, ν_j)+𝒪(^-C t)   if κ_j>0, 
               q_sol(x,t; c_ł, κ_j, x_j)+𝒪(^-C t)   if κ_j=0,
         for some  constant  C>0. Here ν_j  and x_j are  defined in (<ref>).
        
        We thus have finished the proof of Theorem <ref>   part (a).
        
        
        § SUBLEADING  TERM OF ASYMPTOTIC EXPANSION  IN THE LEFT REGIONS Ξ<-C_-^2/2 AND -C_-^2/2<Ξ<-C_-^2/2+C_+^2
        
        
         §.§ Second term of asymptotics in the utmost left constant region ξ<-c_-^2/2 or x<-6c_-^2t
        
        
        The final RH problem in the utmost left constant region ξ<-c_-^2/2 is depicted in  Figure <ref>.
        
        Note that the jump matrices are exponentially close to the identity matrix everywhere except for the stationary points ± k_0, where k_0=√(-ξ-c_-^2/2) and g'(k_0,ξ)=0.
        
        
        
        We will now construct a piecewise matrix-valued function, which satisfies approximately the jumps in the vicinities of the points k=± k_0. Function g(k,ξ).
        We start by inspecting the behavior of the phase function g(k,ξ) in the neighborhoods of the points ± k_0. Note that g is an odd function, g(k)=-g(-k). We have
        
            g(k,ξ)=g(k_0,ξ)+1/2g”(k_0,ξ)(k-k_0)^2+𝒪(k-k_0)^3,   k→ k_0,
        
        where g”(k_0)=-24(ξ+c_-^2/2)/√(c_-^2/2-ξ)>0, and 
        
            g(k,ξ)=-g(k_0,ξ)-1/2g”(k_0,ξ)(k-k_0)^2+𝒪(k-k_0)^3,   k→ -k_0.
        
        
        This prompts us to introduce the local conformal changes of variables,
        
            z^2 =: g(k,ξ)-g(k_0,ξ), ,
                
            z_l^2 =: -g(k,ξ)-g(k_0,ξ), ,
        
        where g(k_0,ξ)=-8-ξ+c_-^2/2^3/2.
        We also introduce the rescaled variables 
        
            ζ:= √(t ) z,  ζ_l:=√(t )  z_l.
        
        Then, using the expansion of the function g(k) near ± k_0 we have 
        
        
            z=√(g”(k_0,ξ)/2)(k-k_0)1+𝒪(k-k_0),  k→ k_0,
            
            
            z_l = -√(g”(k_0,ξ)/2)(k+k_0)1+𝒪(k+k_0), k→-k_0.
        
        
        Note that the  left variable z_l is rotated 180 degrees compared to the orientation of k+k_0, while the right variable z preserves the orientation of k-k_0.
        
         Function T(k).
        Next, we need to understand the behavior of the function T(k)  defined in (<ref>) as k approaches the points ± k_0. Note that the singular behavior of T(k) is described 
         by the function k-k_0/k+k_0^-ıν, where ν=1/2πln(1+|r(k_0)|^2).  We define the function χ(k) in such a way that 
        
            T(k)=:k-k_0/k+k_0^-ıν(ξ)χ(k,ξ),
              ν(ξ)=1/2πln1+|r(k_0)|^2,
        
        and the function χ is meromorphic in ℂ∖[-k_0,k_0]∪[ı c_-,-ı c_-], and has a non-zero limit as k→± k_0 (we thus have that the function χ is continuous at the points ± k_0, though it has discontinuity across the segment [-k_0,k_0]).
        
        Note that the function T(k) possesses the following symmetries:
        
            T(- k) = T(k),  T( k) T(k)=1,
        
        Singling out the main components of the function T(k), we can write it as 
        
            T(k)=ζ^-ıν·k-k_0/  (k+k_0) √(t )z^-ıν·χ(k,ξ),   k→ k_0,
        
            T(k)=ζ_l^ıν·(k-k_0)z_l√(t)/k+k_0^-ıν·χ(k,ξ),  
            k→-k_0.
        
        
          §.§.§ Parabolic cylinder functions
        
        The local solution  near the points ± k_0  of the RH problem in Figure <ref>  is obtained though the Parabolic Cylinder functions.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        The goal of this section  is to construct a matrix function that reproduces in an approximate way the jumps of X(k) in a neighbourhood of ± k_0.
        We follow <cit.> to obtain the local parametrices.
        The Parabolic Cylinder function D_a(z), a∈ℂ, is  an entire function of z,  defined as the solution of the equation
        
            D_a”(z)+(a+1/2-z^2/4)D_a(z)=0,
        
        with the asymptotic behaviour 
        
            D_a(z)=z^a^-1/4z^2(1+𝒪(z^-1))  z→∞ z∈(-3π/4,3π/4).
        
        Furthermore, the parabolic cylinder function satisfies the  relations
        
            D_a(z)=D_a(-z)^-πı a+√(2π)/Γ(-a)^-πı/2(a+1)D_-a-1(ı z),
            
               
            D_a(z)=D_a(-z)^πı a+√(2π)/Γ(-a)^πı/2(a+1)D_-a-1(-ı z).
        
        
        Let r_* and ρ_* be two nonzero  complex numbers such that r_*ρ_*≠ -1 and  denote  by ν=1/2πln1+r_*ρ_*.
        Let us define the function
        
            Ψ(ζ) = 
            [       2^ıν^3πν/4D_-ıν(2^-3πı/4ζ)  β_1 2^-ıν^-πν/4D_ıν-1(2^-πı/4ζ); β_2 2^ıν^3πν/4D_-ıν-1(2^-3πı/4ζ)        2^-ıν^-πν/4D_ıν(2^-πı/4ζ) ],
              ζ>0,
        
            Ψ(ζ)=
            [         2^ıν^-πν/4D_-ıν(2^πı/4ζ) -β_1 2^-ıν^3πν/4D_ıν-1(2^3πı/4ζ);  -β_2 2^ıν^-πν/4D_-ıν-1(2^πı/4ζ)        2^-ıν^3πν/4D_ıν(2^3πı/4ζ) ],
              ζ<0,
        
        where 
        
            β_1=-ı√(2π) 2^2ıν^πν/2/r_* Γ(ıν)
            =
            ρ_* Γ(-ıν+1)2^2ıν/√(2π) ^πν/2,
              β_2=
            r_* Γ(1+ıν)/√(2π)  2^2ıν^πν/2
            =
            ı√(2π) ^πν/2/ρ_*2^2ıνΓ(-ıν),
        
        with β_1β_2=ν.
        The function Ψ(ζ) is a piecewise analytic function   and using the relations (<ref>), it has the following jump over the real axis:
        
            Ψ_-(ζ)=Ψ_+(ζ)[        1     -ρ_*;     -r_* 1+r_*ρ_* ],  ζ∈ℝ,
        
        where Ψ_±(ζ) are the boundary values of Ψ(ζ) as ζ approaches the oriented real line.
        
        
        
        
        
        
        Next we introduce the function Φ(ζ) defined as 
        
            Φ(ζ)=
               Ψ(ζ)ζ^ıνσ_3^ıζ^2σ_3,
              ζ∈(π/4,3π/4)∪(-3π/4,-π/4),
            
               Ψ(ζ)ζ^ıνσ_3^ıζ^2σ_3[               1               0; -r_*ζ^2ıν^2ıζ^2               1 ],  ζ∈(0,π/4),
            
               Ψ(ζ)ζ^ıνσ_3^ıζ^2σ_3[                1 ρ_*ζ^-2ıν^-2ıζ^2;                0                1 ],  ζ∈(-π/4,0),
            
               Ψ(ζ)ζ^ıνσ_3^ıζ^2σ_3[                          1 -ρ_*/1+r_*ρ_*ζ^-2ıν^-2ıζ^2;                          0                          1 ],  ζ∈(3π/4,π),
            
               Ψ(ζ)ζ^ıνσ_3^ıζ^2σ_3[                       1                       0; r_*/1+r_*ρ_*ζ^2ıν^2ıζ^2                       1 ],  ζ∈(-π, -3π/4).
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        The function Φ(ζ) satisfies the RH problem  Φ_-(ζ)=Φ_+(ζ)J_Φ(ζ),
        where
        
        
        
            J_Φ(ζ)
            =[               1               0; -r_*ζ^2ıν^2ıζ^2               1 ],   ζ∈(0,^πı/4∞),
               J_Φ(ζ)=[                 1 -ρ_*ζ^-2ıν^-2ıζ^2;                 0                 1 ],  ζ∈(0,^-πı/4∞),
            
               J_Φ(ζ)=
            [                          1 -ρ_*/1+r_*ρ_*ζ^-2ıν^-2ıζ^2;                          0                          1 ],
              ζ∈(^3πı/4∞,0),
            
               J_Φ(ζ)=
            [                        1                        0; -r_*/1+r_*ρ_*ζ^2ıν^2ıζ^2                        1 ],  ζ∈(^-3πı/4∞,0).
        
        Besides, the function Φ(ζ) has the following uniform asymptotics as ζ→∞,
            Φ(ζ)=
            [    1 -ν(1+ıν)/8ζ^2(1 +ı(2+ıν)(3+ıν)/16ζ^2)   ^πı/4β_1/2ζ
            +^-πı/4β_1(1-ıν)(2-ıν)/16ζ^3; ^3πı/4β_2/2ζ
            +^-3πı/4β_2(1+ıν)(2+ıν)/16ζ^3      1-ν(1-ıν)/8ζ^2(1-ı(2-ıν)(3-ıν)/16ζ^2) ]+𝒪(1/ζ^5).
        
        
          §.§.§ Approximation, far away from breathers
        
        Now we are almost ready to define an approximation for the function X(k). For this  purpose  we define  
          right and  left parametrices
        
            P_r(ζ) = Φ(ζ)|_r_*=r(k_0),ρ_*=r(k_0) ,
                
            P_l(ζ_l)=[  0 -1;  1  0 ]Φ(ζ_l)|_r_*=r(k_0),ρ_*=r(k_0)[  0  1; -1  0 ].
        
        where Φ has been defined in the previous Section <ref>.
        Note that the parameters r_*, ρ_* are the same for both P_r and P_l, and that P_l is additionally shuffled by  off-diagonal matrices.
        
        
        Let  δ>0,  such that the conformal changes of coordinates z, z_l exist inside the disks  
            𝒰_± k_0={k∈  s.t. |k± k_0|≤δ}.
        
        We define the boundaries of the disks 𝒰_± k_0 as ∂𝒰_± k_0 and we assume that they are oriented anticlockwise.
        Furthermore we define 
        
            P^(k_0)(k)=ℬ_r(k)P_r(ζ(k))
            ·^-ı t g(k_0)σ_3χ(k_0)^σ_3(k-k_0)/(k+k_0)z√(t)^ıνσ_3,   k∈𝒰_k_0, 
                P^(-k_0)(k)=ℬ_l(k)P_l(ζ_l(k))
            ·^-ı t g(k_0)σ_3χ(k_0)^σ_3(k-k_0)z_l√(t)/k+k_0^ıνσ_3,   k∈𝒰_-k_0,
        
        where ℬ_r(k) and ℬ_l(k) are matrices that will be determine below.
        We are now ready to define the approximate solution of X(k)  as 
        
            X_appr(k)=
            P^(k_0)(k) 
            
             P^(-k_0)(k) 
            
            M^(mod)(k) ,
        
        where M^(mod)(k)=[ a_γ(k) b_γ(k); b_γ(k) a_γ(k) ],a_γ(k)=1/2(γ(k)+1/γ(k)),b_γ(k)=1/2(γ(k)-1/γ(k)),γ(k)=√(k-ı c_-/k+ı c_-).
        
        The matrices ℬ_r,l(k) are obtained by requiring that 
        
            P^(± k_0)(k)=M^(mod)(k)(I+o(1))   ,
        
         where Σ^±=∪_j=1^4L_j∩𝒰_± k_0.
         This gives 
         
            ℬ_r(k)=M^(mod)(k)^-ı t g(k_0,ξ)σ_3χ(k_0,ξ)^σ_3k-k_0/√(t) z(k+k_0)^-ıνσ_3, 
            ℬ_l(k)=M^(mod)(k)^ı t g(k_0,ξ)σ_3χ(k_0,ξ)^-σ_3(k-k_0)z_l√(t)/k+k_0^-ıνσ_3,  .
        
        We observe that the matrices ℬ_l,r(k) are holomorphic inside the circles 𝒰_± k_0.
        
        
         Next we consider the  error matrix E(k)
            E(k)=X(k)X_appr(k)^-1 .
        
        Such  a matrix has jump J_E(k)=(E_+(k))^-1 E_-(k). It is easy to check that J_E(k) is exponentially close to the identity as t→∞ for k∈\𝒰_± k_0.
        We observe that since P_r,l(ζ)=I+𝒪(t^-1/2) on the circles ∂𝒰_± k_0
        we also have  J_E(k)=I+𝒪(t^-1/2) as t→∞ and k∈∂𝒰_± k_0.
        Finally, let us estimate the jump matrix J_E on contours Σ^±= ∪_j=1^4L_j∩∂𝒰_± k_0.
        The terms which appear in the jump matrix J_E(k)   for k∈Σ^+∪Σ^- 
        are of the form
        
            r(k) - r(k_0)ζ^2ıν^2ıζ^2,
              r(k)/1+r(k)r( k) - r(k_0)/1+|r(k_0)|^2ζ^2ıν^2ıζ^2
        
        and they can be estimated as 
        
            |r(k) - r(k_0)ζ^2ıν^2ıζ^2|
               ≤ C
            max_k∈Σ^±|k-k_0|
            ^-2|ζ|^2
               ≤ C
            max_z≥0|z|
            ^-2t |z|^2
            =
            C/√(t)max_|z|≥0|z|·^-2 |z|^2
         
        with some generic constant C>0 (taking the maximum, we first allowed |z| to vary over the whole positive numbers [0,+∞) instead of a finite interval, and second, we  make the transformation |z|√(t)|z|),
        
        
        and similarly
        
            |r(k)/1+r(k)r( k) - r(k_0)/1+|r(k_0)|^2ζ^2ıν^2ıζ^2|=𝒪(1/√(t)).
        
        
        
        
        
        
        
        We conclude that  jump matrix J_E admits an estimate I+𝒪(t^-1/2) uniformly on the whole contour Σ_E= Σ^±∪∂𝒰^±.
        
        
          §.§.§ Second term of the asymptotics
        
        We thus can look for the  error matrix  E(k) in the form 
        
            E(k)=I+𝒞(F),
        
        where 𝒞 is the Cauchy operator,
        
            𝒞f(k)=1/2πı∫_Σ_Ef(s)ṣ/s-k,
        
        where  Σ_E= Σ^±∪∂𝒰_± k_0.
        Taking the limiting values in (<ref>), and using the relation 𝒞_+-𝒞_-=I, we find F(s)=E_+(s)(I-J_E(s)).
        Thus,
        
            E(k)=I+𝒞[E_+(s)(I-J_E(s))],
              
            E_(k)=I+1/2πı∫_Σ_EE_+(s)(I-J_E(s)) ṣ/s-k,
        
        where E_+ is the solution of the singular integral equation E_+(k)=I+𝒞_+[E_+(s)(I-J_E(s))].
        Define
        
            q_err(x,t):=lim_k→∞2ı k(E(k)-I)_12=lim_k→∞2ı k(E(k)-I)_21,
            
               q_appr(x,t):=lim_k→∞2ı k(X_appr(k)-I)_12=lim_k→∞2ı k(X_appr(k)-I)_21;
        
        then q(x,t)=q_err(x,t)+q_appr(x,t), and q_appr=c_-.
        From (<ref>) we find an expression for q_err,
            q_err(x,t)=1/π∫_Σ_E[E_+(s)(J_E(s)-I)]_21ṣ.
        
        Subtracting and adding the identity matrix from E_+, and observing that since J_E(k)=I+𝒪(t^-1/2) uniformly on the contour Σ_E, then also 
        E(k)=I+𝒪(t^-1/2), we find that 
        
            q_err(x,t)=1/π∫_Σ_E(J_E(s)-I)_21ṣ+𝒪(t^-1).
        
        We start by estimating the integral over the parts of L_j, j=1,2,3,4, inside the circles |k∓ k_0|<δ. They admit an estimate of the type
        
            ∫_0^δy^-2 ty^2ỵ<1/t∫_0^∞(y√(t))^-2ty^2(̣y√(t))=C/t,
        
        where C is a generic constant, and thus do not contribute to the t^-1/2 term.
        The rest of the contour except for the circles around the points ± k_0 gives an exponentially small contribution. Next we compute the integrals over the circles |k∓ k_0|=δ, which we orient in the counter-clock-wise direction. 
        Due to the symmetries E(- k)=E(k)=[  0  1; -1  0 ]E(-k)[  0 -1;  1  0 ], the integrals over the circles ∂𝒰_± k_0   have the following structure:
        
            1/π∫_∂𝒰_k_0(J_E(s)-I)ṣ = [ … A; A … ],
              1/π∫_∂𝒰_-k_0(J_E(s)-I)ṣ = [ … A; A … ],
        
        and hence q_err=2 A+𝒪(t^-1).
        Note that for k∈∂𝒰_k_0 the jump matrix equals
        
            J_E(k)-I=ℬ_r(k)[[         0  ^πı/4β_1; ^3πı/4β_2         0 ]1/2ζ+𝒪(ζ^-2)]
            (ℬ_r(k))^-1
        
        where ℬ_r(k) is defined in (<ref>).
        
        Neglecting the 𝒪(ζ^-2) terms gives an error of the order t^-1, and the terms of the order ζ^-1 have simple a pole at k=k_0, and thus they can be computed by taking residues.
        
        
        
        Since 
        
            |^-ı t g(k_0)χ(k_0) √(t ) 2k_0√(g”(k_0)/2)^ıν|=1,
        
        we denote 
        
            e^ıϑ=^-ı t g(k_0)χ(k_0) √(t ) 2k_0√(g”(k_0)/2)^ıν.
        
        Then
        
            A:=1/π∫_∂𝒰_k_0(J_E(k)-I)_21ḳ = 
            1/π∫_∂𝒰_k_0a_γ(k)^2/e^2ıϑ^3πı/4β_2
            -b_γ(k)^2e^2ıϑ^πı/4β_1ḳ/2ζ
            =
            
               =2ı Res_k=k_0a_γ(k)^2/e^2ıϑ^3πı/4β_2
            -b_γ(k)^2 e^2ıϑ^πı/4β_1ḳ/2ζ
               =
            1/√(t g”(k_0)/2)a_γ(k_0)^2/ e^2ıϑ^-3πı/4β_2
            -b_γ(k_0)^2 e^2ıϑ^3πı/4β_1.
        
        We have q_err(x,t)=A+A+𝒪(t^-1), and thus we need to find the real part of the constant A.
        Substituting the expressions for a_γ(k)=1/2(γ(k)+1/γ(k)),b_γ(k)=1/2(γ(k)-1/γ(k)),γ(k)=√(k-ı c_-/k+ı c_-),
         and taking into account that |γ(k_0)|=1  and β_1=β_2
        we find
        
            A=1/√(t g”(k_0)/2)   {1/2(2k_0/√(k_0^2+c_-^2) )
            [
             e^-2ıϑ^-3πı/4β_2
            - e^2ıϑ^3πı/4β_1
            ]_∈ıℝ+.
               .+
            1/2e^-2ıϑ^-3πı/4β_2
            + e^2ıϑ^3πı/4β_1}.
        
        The underbraced expression is purely imaginary, and hence does not contribute to the real part of A.
        We have
        
            2 A = 2/√(g”(k_0)t/2)e^2ıϑβ_1^3πı/4,
        
        and substituting expressions for β_1 from Section <ref>,
        we can write the latter in more detail,
        
            2    A=
             2/√(g”(k_0)t/2)·√(2π ) ^πν/2/|r(k_0)| |Γ(ıν)|·
               ·cos[
             π/4-(r(k_0)Γ(ıν))-2tg(k_0)+χ^2(k_0)
             +
             2νln4k_0√(tg”(k_0)/2)].
        
        The second factor here can be computed using properties of the Gamma-function, √(2π ) ^πν/2/|r(k_0)| |Γ(ıν)| = √(ν),
            q_err(x,t)=2 A+𝒪(t^-1)
            =√(8ν/t· g”(k_0))×
                  ×cos[π/4
            -2t g(k_0)+νln8 t k_0^2 g”(k_0)
            -(r(k_0)Γ(ıν))+χ^2(k_0)
            ]
            +𝒪(t^-1).
        
        Substituting finally the expressions for g(k_0), g”(k_0), we get 
        
            q_err(x,t)=
            √(ν√(-ξ+c_-^2/2)/3t(-ξ-c_-^2/2))×
                 ×cos[
            16t-ξ+c_-^2/2^3/2
            +νln192  t  (ξ+c_-^2/2)^2/√(-ξ+c_-^2/2)
            +
            ϕ(k_0)
            ]
            +𝒪(t^-1),
        
        where the phase shift is given by the formula 
        
            ϕ(k_0):=π/4- r(k_0)-Γ(ıν)+χ^2(k_0).
        
        Note that the formula (<ref>) is very similar to the radiation part of the rarefaction wave <cit.>, but differs from it by the phase shift π.
        Note also, that the error term is uniform in ξ<-c_-^2/2-ε, for a  positive ε>0. Looking at (<ref>), we see that it blows up on the line ξ=-c_-^2/2, which probably indicates the presence of a nontrivial transition region.
        
        
        
        
        
        
         §.§ Second term of the asymptotic expansion  in the middle left constant region -c_-^2/2<ξ<-c_-^2/2+c_+^2 or -6c_-^2t<x<(-6c_-^2+12c_+^2)t
        
        
        
        
        The jumps for the final RH problem <ref>  for the matrix function X(k)   in the middle left constant region   considered in Section <ref> are depicted in  Figure <ref>.
        
        Note that the jump matrices are exponentially close to the identity matrix everywhere except for interval [ı c_-, -ı c_-],  near  the stationary points ±ı d_0, where d_0=√(ξ+c_-^2/2) where  g'(ı d_0,ξ)=0, where the function g(k,ξ) has been defined in (<ref>).
        
        
        
        We will now construct a piecewise matrix-valued function, which satisfies approximately the jumps in the vicinities of the points k=±ı d_0.
        The analysis is similar to that of Section <ref>, but it is slightly more involved because of discontinuity across the imaginary axis. This however can be handled by multiplication of the matrix by an off-diagonal matrix in the left half-disk around the point ı d_0. We provide analysis for the upper point  ı d_0, since the constrution at the point -ı d_0 follows by symmetry.
        
        
        
        
         Function g(k,ξ).
        We start by inspecting the behavior of the phase function g(k,ξ) in the neighborhoods of the points ±ı d_0. We recall  that 
            g(k,ξ)=22k^2-c_-^2+6ξ)√(k^2+c_-^2) 
                has 
                 a discontinuity across [ı c_-,-ı c_-],   because we chose √(k^2+c_-^2) analytic in \[ı c_-,-ı c_-]. Furthermore, we have the symmetry g(k)=-g(- k). 
                 For δ >0 let
            𝒰_±ı d_0={k∈ , s.t. |k∓ı d_0|<δ}.
                We also define 𝒰_ı d_0^±  the semicircles on the left and right part of the complex plane with respect to the imaginary axis.
                 The expansion of  g(k,ξ) near  the point k=ı d_0 takes the form
                
                    g(k,ξ)=±g_+(ı d_0,ξ)+1/2g”_+(ı d_0,ξ)(k-ı d_0)^2+𝒪(k-ı d_0)^3
                    ,  k→ı d_0, k∈𝒰_ı d_0^±,
                
                where
                
                    g_+(ı d_0) = -8c_-^2/2-ξ^3/2<0,
                         g_+”(ı d_0)=-24(ξ+c_-^2/2)/√(c_-^2/2-ξ)<0.
                
                
                 We  introduce the local conformal changes of variables,
                
                    z:=√(g(k)- g_+(ı d_0))  ,    k∈𝒰_ı d_0^+, 
                    √(-g(k)- g_+(ı d_0))  ,      k∈𝒰_ı d_0^- .
                
                We have that
                
                    z=√(|g”(k_0)|/2)k-ı d_0/-ı1+𝒪(k-ı d_0), k∈𝒰_ı d_0 .
                
                We also introduce
            ζ:= √(t ) z.
                Note that the local variable z is rotated by 90 degrees with respect  to the orientation of k-ı d_0.
                
                Summing up, we have
                
                    ± 2ı t g(k,ξ)=2ı t g_+(ı d_0,ξ)+2 ı t z^2 = 2ı t g(k_0,ξ) + 2ıζ^2, k∈𝒰_ı d_0^±  .
                 Function T(k).
                Next, we need to understand the behavior of the function T(k) as k approaches the point ı d_0.  We recall that the function T satisfies the jump relations 
                T_+(k)T_-(k)=1/|a(k)|^2, k∈(ı c_-,ı d_0), and T_+(k)T_-(k)=1, k∈(ı d_0, -ı d_0), and 
                T_+(k)T_-(k)=1/|a|^2, for k∈(-ı d_0, -ı c_-). To understand the behavior of the function T(k) at the point k=ı d_0 it is convenient to introduce the function 
                
                    F(k)=   T(k),  , 
                       1/T(k),
                
                 Then the function F(k) satisfies the jump F_+(k)/F_-(k)=1/|a(k)|^2 on the interval (ı c_-,ı d_0), and it is continuous across the interval (ı d_0,-ı d_0).
                After straighforward computation we conclude that
                
                    T(k)=   k-ı d_0/-(k+ı d_0)^-ıνχ(k), 
                       k-ı d_0/-(k+ı d_0)^ıν1/χ(k),
                
                and the latter formula is the definition of the  function χ(k), which is discontinuous across [ı c_-,ı d_0]∪ [-ı d_0,-ı c_-] and continuous across [ı d_0,-ı d_0], and has a non-zero limit as k→±ı d_0. Here
                
                    ν=-1/2πln|a(ı d_0)|^2<0.
                
                By Lemma <ref>,  we have that |a(ı d_0)|^2-|b(ı d_0)|^2=1 and hence indeed ν<0.
                Note that |χ(ı d_0)|=1 in view of the symmetry relation 
                T(- k) = T(k).
                Singling out the main components of the function T(k), we can write it as 
                
                    T(k)=ζ^-ıν·k-ı d_0/-(k+ı d_0) √(t )  z^-ıν·χ(k,ξ),  
                    
                       T(k)=ζ^ıν·k-ı d_0/-(k+ı d_0) √(t )  z^ıν·1/χ(k,ξ),
                
                In Figure <ref> we plot the image  of the contours L_1, L_5 and L_7 passing through  the point k=ı d_0 via the map k→ζ(k).  We denoted by L_1r and L_1l the images of the parts of the contour L_1 with k>0 and k<0, respectively.
                
                
                
                
                  §.§.§ Approximation, far away from breathers
                
                Now we are almost ready to define an approximation for the function X(k)  that solves the RH problem <ref> in the middle  left constant region.
                 For this purpose we use the local parametrix Φ(ζ) in terms of parabolic cylinder functions obtained  in section <ref>.
                 We define the  upper parametrix as
                
                    P_u(k)=B(k)  Φ(ζ(k))|_{r_*=r_+(ı d_0), ρ_*=-r_+(ı d_0)}·(^ı t g_+(ı d_0,ξ)/χ(ı d_0))^σ_3-(k-ı d_0)/(k+ı d_0) z √(t)^ıνσ_3,
                
                
                Note that the parameter r_* is the same as  in  Section <ref>, but the parameter ρ_* has the opposite sign compared to the Section <ref>.  The matrix B(k)  in the above expression will be determined below.
                
                
                Now we are ready to define an approximation of the function X(k).  We define
                
                    X_appr(k)=
                    P_u(k),  ,
                    
                    P_u(k)
                    [ 0 ı; ı 0 ],
                      
                    [  0  1; -1  0 ]P_u( k)[  0 -1;  1  0 ],
                     
                    
                    -ı[  0  1; -1  0 ]P_u( k)σ_3,
                     
                    
                    M^(mod)(k),   |k∓ı d_0|>δ,
                
                
                
                where M^(mod)(k)=[ a_γ(k) b_γ(k); b_γ(k) a_γ(k) ],a_γ(k)=1/2(γ(k)+1/γ(k)),b_γ(k)=1/2(γ(k)-1/γ(k)),γ(k)=√(k-ı c_-/k+ı c_-).
                The matrix B(k) in (<ref>) is obtained by requesting that   the error matrix
                 
                    E(k)=X(k)X_appr(k)^-1
                
                has its jumps 
                as close to the identity matrix  as possible. In particular for k∈∂𝒰^+_ı d_0  we  have J_E(k)=E^-1_+(k)E_-(k)=(P_u(k))^-1M^(mod)(k) where we assume that the circle 
                𝒰_ı d_0 is oriented anticlockwise.
                Requesting that J_E=I+o(1) we obtain 
                
                    B(k)=   M^(mod)(k)(χ(ı d_0,ξ)/^ı t g_+(ı d_0,ξ))^σ_3-(k-ı d_0)/(k+ı d_0) z √(t )^-ıνσ_3,   k∈𝒰^+_ı d_0, 
                       M^(mod)(k)[ 0 ı; ı 0 ]^-1(χ(ı d_0,ξ)/^ı t g_+(ı d_0,ξ))^σ_3-(k-ı d_0)/(k+ı d_0) z √(t )^-ıνσ_3,   k∈𝒰^-_ı d_0.
                
                We observe that with the above definition the matrix B(k) is analytic in 𝒰_ı d_0.
                
                
                In a similar  way as  in Section <ref>, we can verify that the jump matrix J_E(k)=E^-1_+(k)E_-(k) 
                is of the order J_E(k)=I+𝒪(t^-1/2) uniformly for k∈∂𝒰_±ı d_0 and on the contours 
                (L_1∪ L_5∪ L_7)∩𝒰_ı d_0 and (L_2∪ L_6∪ L_8)∩𝒰_-ı d_0,
                 while on the remaining contours the jump matrix J_E is exponentially close to the identity matrix.
                 Only the circles ∂𝒰_±ı d_0 contribute to the t^-1/2-term of the asymptotic expansion  of the MKdV solution q(x,t), while the remaining terms give a  higher order  contribution.
                 We thus have 
                
                    q(x,t)=q_appr(x,t)+ q_err(x,t),
                
                where
                
                    q_err(x,t)=1/π∫_𝒰_ı d_0(J_E(k)-I)_21ḳ + 1/π∫_𝒰_-ı d_0(J_E(k)-I)_21ḳ
                    +𝒪(t^-1).
                
                The symmetries E(- k)=E(k)=[  0 -1;  1  0 ]
                E(-k)=E( k)[  0  1; -1  0 ]
                imply that the integrals over the oriented circles ∂𝒰_±ı d_0 have the following structure as t→∞:
                
                    1/π∫_∂𝒰_ı d_0(J_E(k)-I)ḳ
                    =[ … B; A … ],
                        1/π∫_∂𝒰_- ı d_0(J_E(k)-I)ḳ
                    =[ … A; B … ],
                
                and both A,B are real.
                Hence, q_err=A+B+𝒪(t^-1), and it suffices to make   the computation only on the cirlcle ∂𝒰_ı d_0.
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                For simplifying the notation we define
                
                    f    := ^-ı t g_+(ı d_0)χ(ı d_0) lim_k→ı d_0k-ı d_0/-(k+ı d_0)z√(t)^-ıν
                       =^-ı t g_+(ı d_0)χ(ı d_0) 2 d_0√(|g”_+(ı d_0)| t/2)^ıν,
                
                and observe that |f|=1.   Then the integral takes the form
                
                    ∫_∂𝒰_ı d_0(J_E(k)-I)_12ḳ =^πı/4√(2π^2/|g”_+(ı d_0)| t)(a_γ,+(ı d_0)^2f^2β_1-ıb_γ,+(ı d_0)^2/f^2β_2)+𝒪(1/t),
                       ∫_∂𝒰_ı d_0(J_E(k)-I)_21ḳ =^3πı/4√(2π^2/|g”_+(ı d_0)| t)(a_γ,+(ı d_0)^2/f^2β_2+ı
                    b_γ,+(ı d_0)^2f^2β_1)
                    +𝒪(1/t) .
                
                We can use the relations β_2=-β_1 and a_γ,+(ı d_0) =ı b_γ,+(ı d_0),b_γ,+(ı d_0) =ı a_γ,+(ı d_0)
                to verify that indeed A,B∈ℝ.
                We have 
                q_err(x,t)=A+B+𝒪(t^-1), and using the relation a_γ,+(ı d_0)^2-b_γ,ı(ı d_0)^2=1, we obtain, after some simplifications,
                
                    q_err(x,t)   =√(2/|g”_+(ı d_0)|  t)^πı/4 β_1 f^2
                    +^-πı/4β_1/f^2+
                    𝒪(t^-1)
                       =
                    √(2/|g”_+(ı d_0)|  t)· 2^πı/4 β_1f^2+𝒪(t^-1).
                
                Substituting the expression for f,g_+(ı d_0) and g”_+(ı d_0)  
                
                
                
                
                
                
                
                
                
                
                
                
                and the  relation
                
                    √(2π) ^πν/2/|r_+(ı d_0)| |Γ(ıν)|=√(|ν|),
                
                
                
                in the expression for  q_err(x,t),  we   finally obtain 
                
                    q_err(x,t)
                    =√(|ν|√(c_-^2/2-ξ)/3(ξ+c_-^2/2)t)   cos16tc_-^2/2-ξ^3/2+νln192t(ξ+c_-^2/2)^2/√(c_-^2/2-ξ)
                    +ϕ(k_0)+
                                 +𝒪(t^-1) ,
                
                where the phase shift is given by the formula
                
                    ϕ(k_0)=-π/4- r_+(ı d_0)-Γ(ıν)+χ^2(ı d_0).
                
                Note that the formula (<ref>) is very similar to the formula (<ref>), but the term π/4 enters with the opposite sign in the phase shift, and hence the phase shift in formula (<ref>) differs by π/2 from the one in (<ref>).
                
                
                
                
                §  TRAVELLING WAVE SOLUTION OF MKDV  AND WHITHAM MODULATION EQUATIONS
                
                 We look for the solution  of the  MKdV equation
                
                    u_t+6u^2u_x+u_xxx=0
                
                 in the form 
                    u(x,t)=η(θ),   where θ =kx-ω t,  𝒱:=ω/k .
                
                Substituting this ansatz into MKdV equation, we get after two integrations
                
                    k^2η_θ^2=-η^4+vη^2+Bη+A,
                
                where A,B are constants of integration.
                We assume that the above polynomials has four real roots   e_1<e_2<e_3<e_4,   so that
                
                    k^2   η_θ^2=-(η-e_1)(η-e_2)(η-e_3)(η-e_4),
                        ∑_j=1^4e_j=0,
                        𝒱=-∑_i<je_ie_j.
                
                Finite real valued periodic motion can take place   when η varies between [e_1,e_2]  or [e_3,e_4]. We develop the calculations   for the latter case,
                the first  case can be derived by the symmetry η→ -η and e_j→ -e_5-j.
                We obtain the integral 
                
                    k∫^η_e_3%̣ṣη/√((e_1-η)(η-e_2)(η-e_3)(η-e_4))=θ.
                
                After integration we arrive to the expression
                
                    u_per(x,t)=η(θ)=e_1+e_3-e_1/1-e_4-e_3/e_4-e_1cn^2√((e_3-e_1)(e_4-e_2)) /2 (x-vt)|m,
                
                where the function   cn(z|m) is the Jacobi elliptic function of modulus 
                
                    m^2=(e_3-e_2)(e_4-e_1)(e_4-e_2)(e_3-e_1).
                
                Since e_1+e_2+e_3+e_4=0 we make a change of variables {e_1,e_2,e_3,e_4}→{β_1,β_2,β_3}
                defined as  
                    e_1=-β_1-β_2-β_3,   e_2=β_1+β_2-β_3,   e_3=β_1+β_3-β_2,   e_4=β_2+β_3-β_1,
                
                with β_3>β_2>β_1.
                The inverse transformation is 
                
                    β_1=e_2+e_3/2=-e_1+e_4/2,  β_2=e_2+e_4/2=-e_1+e_3/2,  β_3=e_3+e_4/2=-e_1+e_2/2.
                
                
                 This choice of variables relates the elliptic solution to the   band spectrum of the  Zakharov-Shabat  linear operator (<ref>).
                In this way we obtain  the expression 
                
                    u_per(x,t)=-β_1-β_2-β_3+2(β_2+β_3)(β_1+β_3)/β_2+β_3-(β_2-β_1)cn^2√(β_3^2-β_1^2) (x-𝒱t)+x_0|m,
                
                
                where β_3>β_2>β_1,  the speed 𝒱=2(β_1^2+β_2^2+β_3^2) and x_0 is an arbitrary phase. The elliptic modulus m  transforms to  
                
                
                    m^2=β_2^2-β_1^2β_3^2-β_1^2.
                
                When the periodic motion takes place between e_1 and e_2 the quantity β_j→β̃_j with β̃_1=-β_1, β̃_2=β_2 and β̃_3=β_3.
                
                 The limiting case of the travelling wave  solutions are 
                 
                  (a)e_1→ e_2  which means  β_1→-β_2  or m→ 0;
                 
                  (b)e_3→ e_4 which means β_2→β_1  or m→ 0;
                
                  (c)e_2→ e_3 which implies β_2→β_3 or m→ 1.
                 
                In the case (a) and (b) we have that cn(z|m)→cos z   as m→0. Parametrizing β_2=β+δ and β_1=β-δ  the periodic solution (<ref>)  in case (b) 
                 has an expansion of the form
                
                    u_per(x,t)=β_3-2δ+4δcos^2√(β_3^2-β^2) (x-(4β^3+2β_3t)+x_0+O(δ^2).
                
                which is the small amplitude limit.
                In  case (a)  and in  the limit β_1→ -β_2=-β we obtain
                
                    u_per(x,t)≃-β_3+2(β_3^2-β^2)/β+β_3-2βcos^2√(β_3^2-β^2) (x-(4β^2+2β_3)t)+x_0|m,
                
                which is a  trigonometric solution.
                The soliton limit is  case (c) when m→ 1 or β_3=β_2=β, and cn(z|m)→(z)
                so that 
                
                    u_per(x,t)   ≃β_1+2β(β+β_1)β (2√(β^2-β_1^2) (x-(4β^2+2β_1)t)+x_0)+β_1,
                
                which coincides with the one-soliton solution  (<ref>)  on the constant background  by identifying β with the spectral parameter κ_0 and β_1 with the constant background c.
                
                
                § WHITHAM MODULATION EQUATIONS
                
                The  Whitham modulation equations are the   slow modulations of the wave parameters β_j=β_j(X,T) of the travelling  wave solution (<ref>), where X=ε x and T=ε t, where ε is a small parameter.
                These equations were first derived by Whitham <cit.> for the KdV case using the method of averaging over conservation laws.
                The same equations  can be  obtained requiring that the travelling wave solution with slow variation of the  wave 
                parameters  satisfies the KdV equation up to an error of order O(ε). 
                After  Whitham's work the  modulations equations have been obtained for a large set of partial differential equations. In particular for the MKdV equation   they have been obtained  by Driscoll and O'Neil <cit.> using the original Whitham averaging procedure.
                The Whitham modulation equations for MDKV wave parameters β_3>β_2>β_1 take the form
                
                    ∂∂ Tβ_j+W_j(β_1,β_2,β_3)∂∂ Xβ_j=0,   j=1,2,3,
                
                where the speeds W_j=W_j(β_1,β_2,β_3) are 
                
                    W_j(β_1,β_2,β_3)=2(β_1^2+β_2^2+β_3^2)+4∏_k≠ j(β_j^2-β_k^2)β_j^2+α,
                       α=-β^2_3+(β_3^2-β_1^2)E(m)K(m),   m^2=β_2^2
                    -β_1^2β_3^2-β_1^2,
                
                where E(m)=∫_0^π/2√(1-m^2sinψ^2)dψ is the complete elliptic integral of the second kind.
                At the boundary  one has 
                
                  * m→ 0 or β_2= ±β_1,   then  W_1(β_1,β_1,β_3)=W_2(β_1,β_1,β_3)=-6β_3^2+12β_1^2 and W_3(β_1,β_1,β_3)=6β_3^3;
                
                  * m→ 1 or β_2=±β_3, then  W_2(β_1,β_3,β_3)=W_3(β_1,β_3,β_3)=4β_3^2+2β_1^2 and W_1(β_1,β_3,β_3)=6β_1^2.
                
                The solution of  Whitham modulation equations   is  obtained as a boundary value problem, namely for an initial monotone decreasing initial 
                data q_0(X), the solution satisfies the boundary conditions
                
                  *  when β_1=β_2, then β_3(X,T)=q(X,T)   that solves the equation q_T+6q^2q_X=0 with initial data q_0(X);
                
                  *   when β_2=β_3 then β_1(X,T)=q(X,T)   that solves the equation q_T+6q^2q_X=0 with initial data q_0(X).
                
                We observe that the Whitham modulation equations for MKdV can be obtained from the Whitham modulation equations for KdV  with speeds V_j(r_1,r_2,r_3) as 
                
                    W_j(β_1,β_2,β_3)=V_j(β_1^2,β_2^2,β_3^2),   j=1,2,3.
                
                
                It was shown in <cit.>  that for r_3>r_2>r_1  one has 
                
                  *  strict  hyperbolicity:   V_3(r_1,r_2,r_3)>V_2(r_1,r_2,r_3)>V_1(r_1,r_2,r_3),
                  *  genuine nonlinearity:  ∂∂ r_jV_j(r_1,r_2,r_3)>0.
                
                It follows that the Whitham equations for MKdV are strictly hyperbolic and genuinely nonlinear only  when all  β_3>β_2>β_1>0.
                Indeed one has 
                
                    ∂∂β_jW_j(β_1,β_2,β_3)=2β_j∂∂ r_jV_j(r_1,r_2,r_3)|_r_k=β_k^2,
                
                so that when one of the β_j=0  the equations are not genuinely nonlinear.
                 
                In particular, if we consider the self-similar solution β_j(X,T)=β_j(z) with z=X/T=x/t we have
                that the  Whitham equations reduce to the form
                
                    (W_j-z)∂∂ zβ_j=0,   j=1,2,3.
                
                The  initial data that corresponds to such  a self-similar solution is the  step initial data 
                
                    q_0(X)=c_+     as  
                    X>0
                    c_-     as    X<0.
                
                We observe that such initial data is invariant under the scaling X→ε x so that 
                we can also consider the initial data q_0(x) in the fast variable x.
                The   self-similar  Whitham equations (<ref>) can be uniquely solved  in the form 
                
                    β_3=c_-,   β_1=c_+,  XT=xt=W_2(c_+,β_2,c_-),   c_-≤β_2≤ c_+ ,
                
                as long as we can invert β_2 as a function of x/t within the region 
                
                    =-6c_+^2+12c_-^2=W_2(c_+,c_+,c_-)< x/t <W_3(c_+,c_-,c_-)=4c_-^2+2c_+^2.
                
                This happens only when W_2(c_+,β_2,c_-) is a strictly monotone function of β_2, that is when β_2>0, namely 
                 when c_->c_+>0.
                Thus we have obtained the solution appearing in the long-time asymptotic regime for  the Cauchy problem for MKdV considered in this manuscript.
                
                When c_->-c_+>0  the solution  of the MKdV  will still  develop a  dispersive shock wave, but the Whitham equations are not in general strictly hyperbolic and genuinely nonlinear. In this case   it can be shown that  the solution of  (<ref>)   is obtained as follows. We define z^*=W_1(c_+,-c_+,c_-)=W_2(c_+,-c_+,c_-)=-6c_-^2+12c_+^2.
                Then for z^*≤x/t≤ W_3(c_+,c_-,c_-)  we have that the solution of (<ref>) with the initial data (<ref>)  is obtained as 
                
                    β_3=c_-,   β_1=c_+,  XT=xt=W_2(c_+,β_2,c_-),
                
                which is solvable for β_2=β_2(x/t) in view of the genuinely nonlinearity.
                For -6c_-^2=W_1(0,0,c_-)  ≤x/t≤ z^*=-6c_-^2+12c_+^2  we have 
                
                    β_3=c_-,   XT=xt=W_1(β_1,β_2,c_-),  XT=xt=W_2(β_1,β_2,c_-).
                
                A  plot of the solution is shown in Figure <ref>.
                   
                
                
                The proof of the solvability of the above system of equations   for β_1=β_1(x/t) and β_2=β_2(x/t) 
                requires further analysis  that has been considered in  a similar setting
                 in <cit.>   and also in <cit.> for the  modulation equations <cit.> for the   Camassa-Holm equation. For nonstrictly hyperbolic equations see also <cit.>.
                A discussion of the   various cases arising in the long-time asymptotics for the  MKdV  solution 
                can be found in <cit.>  A review  of dispersive shock waves for KdV can be found in <cit.>.  
                
                
                
                
                §  PROOFS OF PROPERTIES OF JOST SOLUTIONS
                
                Here we prove Lemma <ref>. For the inspiration, we used the work <cit.>, but our functional spaces are a bit different. Namely, in order to be able to tackle not only continuous, but also discontinuous initial functions q_0(x), like q_0(x)=c_±, for ± x>0, we work with functions of bounded variations, BV_loc(ℝ), rather than merely continuous functions. Such functions admit representation
                
                    q_0(x)=_ _-3mm a.e.q_ac(x)+∑_j α_j χ_(-∞, x_j)(x),
                
                 where the set {x_j} is at most countable,  ∑_j |α_j| =:α<∞,
                and χ_A(x) is the indicator function of a set A. Furthermore, q_ac is an absolutely continuous function, i.e.
                
                    q_ac(x)=q_ac(x_0)+∫^x_x_0q'_ac(x)%̣ṣx,     q_ac'(.)∈ L^1_loc(ℝ).
                 It follows that q_0(x) is locally bounded.
                Furthermore, the associated distributional derivative is the signed measure q̣_0(x),
                    q̣_0(x)=q'_ac(x)x̣-∑_j α_j δ(x-x_j),
                
                where δ is the Dirac δ-distribution (a.k.a. Dirac δ-function).
                For the readers who would like to restrict themselves only to smooth (absolutely continuous) q_0(x), we recommend to think α_j≡ 0, in which case q̣_0(x) = q_0'(x) x̣ is the usual derivative.
                We have the following proposition
                
                
                  (a) Let q_0(x)∈ BV_loc(ℝ), and 
                
                    ∫_-∞^0|x| |q_0(x)-c_-|x̣+ ∫^+∞_0|x| |q_0(x)-c_+|x̣ < ∞,
                
                and esssup_-5mm x∈ℝ|q_0(x)|<∞.
                
                Then there are exist solutions Φ^±(x;k) of x-equation (<ref>), such that
                
                    Φ^±(x;k) = E^±(x;k) + ∫_±∞ ^ x L^±(x,y) E^±(y;k) ỵ,
                        k∈ℝ∪(ı c_±,-ı c_±),
                
                where 
                
                    L^±(x,y)=[  L_1^±(x,y)  L_2^±(x,y); -L_2^±(x,y)  L_1^±(x,y) ]
                
                and 
                ∓∫_±∞ ^ x |L_j^±(x,y)| ỵ <∞, where j=1,2. 
                Moreover, L^±(x,y) admit the following estimates in terms of the auxiliary quantities
                
                    σ̂^±(x) =    ∓∫_±∞ ^ x |q_0(x)-c_±|%̣ṣx≥ 0,
                      σ̂_1^±(x) = ∓∫_±∞ ^ x σ̂^±(x) %̣ṣx≥ 0,
                       M^±(x) = 
                    esssup_-5mmu∈(±∞, x)|q_0(u)|,
                
                which exist in view of (<ref>).
                Then  
                
                    |L^±(x,y)| ≤ C[M^±(x), σ̂_1^±(x) ] ·σ̂^±(x+y/2),
                
                where C[M^±(x), σ̂_1^±(x) ] is a constant, which depend only on M^±(x),σ̂_1^±(x), and whose explicit form is not important for us.
                
                  (b) Let in addition to assumptions (a)  suppose that 
                
                    ∓∫_±∞^0|q̣_0(x)|<∞,
                        ∓∫_±∞^0 esssup_-5mm x∈(±∞, x)|q_0(x)| x̣<∞
                
                Then (we supress ± everywhere for the ease of reading)
                
                    Φ(x;k)    = E(x;k) + 1/ı k(q_0(x)-c)σ_1 E(x;k)-
                    1/ı k∫_∞^x σ_1 E(y;k)_̣yq_0(x+y/2)-
                       
                    -L(x,x)σ_3E(x;k)/ı k-
                    1/ı k∫_∞^x q_0(x+y/2)-cσ_1 Q_c E(y;k)ỵ+
                       +1/ı k∫_∞^xL_y(x,y)σ_3 + Lσ_3Q_cE(y;k)ỵ,
                
                where σ_3=[  1  0;  0 -1 ],σ_1=[ 0 1; 1 0 ],Q_c=[  0  c; -c  0 ],
                and 
                    | ∫_∞^x |L_y(x,y)|ỵ|<∞,
                      |∫_∞^x |L(x,y)|ỵ|<∞.
                
                Moreover, (below j=1 or j=2)
                
                    |∂_y L_j(x,y)|≤ C[M(x), σ̂_1(x), σ̂_2(x)]·σ̂(x+y/2) + esssup_-5mmx∈(∞, x)|q_0(x)|,
                
                where C[M(x), σ̂_1(x), σ̂_2(x) ] is a constant, which depend only on M^±(x),σ̂_1(x), σ̂_2(x) and whose explicit form is not important for us, and σ̂_2(x) stands for σ̂_2^±(x)  where 
                
                    σ̂_2^±(x) := ∓∫_±∞^x|q̣_0(x)|≥ 0.
                
                  (c) Let in addition to assumptions (a) and  (b) suppose that 
                
                    ±∫_∓∞^0 ^C|x| |q_0(x)-c| x̣<∞,
                
                with some C>0. Then
                    ±∫_∓∞^0^ C|x| |L(x,y)|ỵ<∞,   ±∫_∓∞^0^ C|x| |L_y(x,y)|ỵ<∞.
                
                The second of conditions (a) follows from the first of conditions (b).
                
                Conditions (b) of the Lemma are satisfied if, for example,
                
                    ∓∫_±∞^0|x| |q̣_0(x)|<∞.
                
                Both conditions (a), (b) of the Lemma are satisfied if, for example,
                
                    ∓∫_±∞^0|x|^2 |q̣_0(x)|<∞ q_0(x) = c_-+∫_-∞^x q̣_0(y) = 
                    c_+-∫^+∞_x q̣_0(y).
                 We start the proof of Proposition <ref>.
                With a bit of abuse of notations, we will suppress the superscript and subscript  ±.
                Assume first that q_0 is an absolutely continuous function. Then one can check directly that
                
                    Φ(x,k)=E(x,k)+∫_∞^xL(x,y)E(y,k)ỵ
                
                is a solution to (<ref>) provided that
                the kernel
                
                    L(x,y)=[  L_1(x,y)  L_2(x,y); -L_2(x,y)  L_1(x,y) ]
                
                satisfies the following system of equations:
                
                
                    L_1,x(x,y)+L_1,y(x,y)=-(q_0(x)+c)L_2(x,y),   y∈(x,∞),
                    
                      L_2, x(x,y)-L_2, y(x,y)=(q_0(x)-c)L_1(x,y),   y∈(x,∞),
                    
                    
                    L_2(x,x)=(q_0(x)-c)/2,  lim_y→∞L_2(x,y)=0.
                
                In its turn, solutions of (<ref>) can be found as solutions of appropriate integral equations. Namely, denote 
                
                    u=x+y/2,   v=x-y/2,     H_1(u,v)≡ L_1(x,y),   H_2(u,v)≡ L_2(x,y),
                
                then we come to the integral equations
                
                    H_1(u,v)=-∫_∞^u(q_0(u+v)+c)H_2(u,v)%̣ṣu,
                       H_2(u,v)=(q_0(u)-c)/2+∫_0^v(q_0(u+v)-c)H_1(u, v)%̣ṣv,
                
                or 
                
                    H_1(u,v)=-1/2∫_∞^u(q_0(u+v)+c)(q_0(u)-c)%̣ṣu-
                               -∫_∞^u(q_0(u+v)+c)∫_0^v(q_0(u+v)-c)
                    H_1(u,v)%̣ṣv%̣ṣu,
                    
                       
                    H_2(u,v)=(q_0(u)-c)/2-∫_0^v(q_0(u+v)-c)∫_∞^u(q_0(u+v)+c)
                    H_2(u,v)%̣ṣu%̣ṣv.
                
                We see that H_2(x,y) has the same level of regularity as q_0 does. Hence, system (<ref>) is appropriate for absolutely continuous functions q_0, but it should have another meaning if we would switch to discontinuous functions q_0 of bounded variations.
                One might think that now we will need to operate with functions of bounded variations of two variables. However, in our case functions of bounded variations of one variable suffices. Indeed, subtracting function q_0(u+v), we obtain a regular function.
                To this end, denote 
                
                    L_1(x,y) = L_1(x,y),  L_2(x,y)=L_2(x,y)-1/2q(x+y/2)-c,
                    
                       H_1(u,v)=H_1(u,v),  H_2(u,v) = H_2(u,v) -1/2(q(u)-c).
                
                Furthermore, H_1, H_2 satisfy the following system of  integral equations:
                
                    H_1(u,v)=-1/2∫_∞^u(q_0(u)-c)(q_0(u+v)+c)%̣ṣu-∫_∞^u(q_0(u+v)+c)H_2(u,v)%̣ṣu,
                       H_2(u,v)=∫_0^v(q_0(u+v)-c)H_1(u, v)%̣ṣv,
                
                or 
                
                    H_1(u,v)=   -1/2∫_∞^u(q_0(u+v)+c)(q_0(u)-c)%̣ṣu-
                       -∫_∞^u(q_0(u+v)+c)∫_0^v(q_0(u+v)-c)
                    H_1(u,v)%̣ṣv%̣ṣu,
                    
                    H_2(u,v)=   -1/2∫_0^v(q_0(u+v)-c)∫_∞^u(q_0(u+v)+c)
                    (q_0(u)-c)%̣ṣu%̣ṣv
                    
                       -∫_0^v(q_0(u+v)-c)∫_∞^u(q_0(u+v)+c)
                    H_2(u,v)%̣ṣu%̣ṣv.
                
                We see that by subtracting from L_2 the `irregular part', which is q_0,  we gain in regularity of L_2. Now L_1, L_2 have one more level of regularity than q_0, and hence we are able to integrate the integral representation for Φ by parts, which gives us k^-1 term in the large k asymptotic expansion.
                
                More precisely, apply the successive approximation method to the first of equations (<ref>),
                i.e. represent H_1 = ∑_j=0^∞H_1^(j),
                where H_1^(0) is the inhomogeneous part of the r.h.s. of the equation, and H_1^(j) is the homogeneous part of the r.h.s. of the equation, applied to H_1^(j-1). Then we have by induction that
                
                    |H_1^j|≤M(u+v)^j+1·σ̂(u) · (σ̂_1(u+v)-σ̂_1(u))^j/2· j!,
                
                where
                
                    σ̂(u)=|∫_∞^u|q(u)-c|%̣ṣu|,
                      σ̂_1(u)=|∫_∞^u σ̂(u) %̣ṣu| = |∫_∞^u(u-u)|q_0(u)|%̣ṣu|,
                
                and M(z) = esssup_-5mmu∈ (∞,z)|q_0(u)+c|.
                Indeed, the induction step goes as follows:
                
                    |H_1^(j+1)(u,v)| ≤|∫_∞^u(q_0(u+v)+c)∫_0^v(q_0(u+v)-c)
                    H_1^(j)(u,v)%̣ṣv%̣ṣu|
                    
                       ≤|∫_∞^u|q_0(u+v)+c|∫_0^v|q_0(u+v)-c|
                    M(u+v)^j+1/2· j!σ̂(u)σ̂_1(u+v)-σ̂_1(u)^j%̣ṣv%̣ṣu|.
                
                In the latter we estimate 
                σ̂(u)≤σ̂(u),  σ̂_1(u+v)-σ̂_1(u)≤σ̂_1(u+v)-σ̂_1(u),    M(u+v)≤ M(u+v),   |q_0(u+v)+c|≤^^^-3mm a.e. M(u+v),
                and thus obtain
                
                    |H_1^(j+1)(u,v)|
                    ≤M(u+v)^j+2/2· j!·σ̂(u)
                    |∫_∞^u σ̂_1(u+v)-σ̂_1(u)^j∫_0^v|q_0(u+v)-c|
                    %̣ṣv%̣ṣu|.
                
                Here we use the definition of σ̂ as the integral of |q-c|, and further make use of |σ̂_1'(u)|=σ̂(u), and integrating by parts, obtain the induction step.
                
                Hence, 
                
                    |H_1(u,v)|≤ K_1(u,v) := 1/2M(u+v)σ̂(u)exp[M(u+v)·(σ̂_1(u+v)-σ̂_1(u))],
                
                and similarly,
                
                    |H_2(u,v)|≤ K_2(u,v)   := 1/2M(u+v) σ̂(u) (σ̂(u+v)-σ̂(u))×
                       ×exp[M(u+v) (σ̂_1(u+v)-σ̂_1(u))],
                
                and henceforth,
                
                    |L_1(x,y)|≤Cσ̂(x+y/2),   |L_2(x,y)|≤Cσ̂(x+y/2),
                
                where C is a generic constant, which does not depend on y. Since σ̂∈ L_1, we henceforth proved the statement (a) of the Lemma.
                
                Passing to the part (b), it is sufficient to prove that |∫_∞^ x |L_y(x,y)|ỵ|<∞, and for this it is sufficient to estimate ∂_uH_1,∂_vH_1,∂_uH_2,∂_vH_2.
                We have 
                
                    ∂_u H_1(u,v) = -(q_0(u+v)+c){H_2(u,v) + 1/2(q_0(u)-c)},
                
                    ∂_v H_2(u,v) = (q_0(u+v)-c) H_1(u,v),
                
                and hence it is enough to estimate ∂_vH_1,∂_uH_2.
                We have 
                
                    ∂_vH_1(u,v)=∫_∞^u [ q_0(u)-c/-2-H_2(u, v)] _̣u q_0(u+v)
                    -∫_∞^u(q_0(u+v)+c) ∂_v H_2(u,v)%̣ṣu, 
                       ∂_u H_2(u,v)=∫_0^v(q_0(u+v)-c)∂_u H_1(u, v)%̣ṣv
                    +
                    ∫_0^v H_1(u, v) _̣v q_0(u+v) ,
                
                and to estimate, we split q̣_0(z) = q_ac'(z)ẓ+ ∑_jα_j δ(z-x_j).
                We thus have 
                
                    |∂_v H_1(u,v)| ≤   σ̂_2(u+v) 1/2esssup_-5mmu∈ (∞, u) |q_0(u)-c| +
                    K_2(u,v) +
                       + M(u+v)  σ̂(u+v)   K_1(u,v),
                    
                     
                    |∂_u H_2(u, v)| ≤    M(u+v)  (σ̂(u+v)-σ̂(u))
                    1/2esssup_-5mmu∈ (∞, u) |q_0(u)-c| + K_2(u,v)+ 
                     
                        + σ̂_2(u+v)-σ̂_2(u) K_1(u, v),
                
                Since 
                    |∫_∞^0 esssup_-5mmu∈ (∞, u) |q_0(u)-c| ụ|  + 
                    |∫_∞^0σ̂(u) ụ| <∞,
                
                we obtain the statement of (b).
                The statement (c) follows immediately from the statement (b)
                and from the estimate (let us stick to -∞ for certainty)
                
                    ∫_-∞^xσ̂(y)^C|y|ỵ    =  
                    ∫_-∞^x^C|y|∫_-∞^y|q_0(z)-c|ẓỵ = 
                       =
                    ∫_-∞^xẓ  |q_0(z)-c| ∫^x_z ^C|y|ỵ = 
                    1/C∫_-∞^x  |q_0(z)-c|  ^C|x| - ^C|z|ẓ.
                
                  (a)
                Provided that conditions (<ref>) are satisfied, the first column Φ^ł_1 is analytic in 
                √(k^2+c_ł^2)>0, i.e. in {k:  k>0}∖[ı c_-,0], the second column Φ^ł_2 is analytic in √(k^2+c_ł^2)<0, and the first column of the right Jost solution Φ^_1
                is analytic in √(k^2+c_^2)<0, and the second column Φ^_2 is analytic in √(k^2+c_^2)>0.
                  (b) Furthermore, define the transition matrix
                T(k):= Φ^+(x;k)^-1Φ^-(x;k), and denote its elements by
                
                    T(k)=[   a(k) -b( k);   b(k)  a( k) ],     r(k) = b(k)/a(k).
                
                Then a(.) is analytic in k∈{k:  k>0}∖ [ı c_-,0], continuous up to the boundary with the exception of the points k∈{ı c_-, ı c_+, 0},
                and has uniform w.r.t. k ∈[0,π] asymptotics a(k)=1+𝒪(k^-1) as k→∞.
                Function r(k) is defined and continuous for k∈ℝ∖{ 0 }, and has the asymptotics
                b(k)=𝒪(k^-1) as k→±∞,k∈ℝ.
                  (c) Suppose that in addition to (<ref>) the following conditions are satisfied: c_-≥ c_+≥ 0, and 
                 
                    ∫_-∞^0|q_0(x)-c_ł|^2|x|√((c_ł+δ)^2-c_ł^2)x̣+∫_0^+∞|q_0(x)-c_|^2x√((c_ł+δ)^2-c_^2)x̣<∞,
                
                for some δ>0. 
                
                Then the Jost solutions Φ^ł,Φ^ are analytic in a δ-neighborhood of the contour k∈Σ=ℝ∪[ı c_ł,-ı c_ł].
                  (d) Furthermore, a(k) and b(k) have analytic extension to k∈ U_δ(Σ)∖ [ı c_-, -ı c_-], and have the asymptotics
                b(k) = 𝒪(k^-1), as k→∞,k∈ U_δ(Σ).
                The  part (a)  of the corollary follows from the fact that under (<ref>) the kernels in (<ref>) are summable:
                
                    ∫_-∞^x|L^ł(x,y)|ỵ<∞,    ∫^+∞_x|L^(̊x,y)|ỵ<∞.
                
                 Statement (c) of the corollary follows directly from the estimates (<ref>) and the fact that under (<ref>)
                    ∫_-∞^0 ^2|y| √((c_ł+δ)^2-c_ł^2)σ̂^ł(y)ỵ<∞,
                        ∫^+∞_0 ^2y √((c_ł+δ)^2-c_^2)σ̂^(̊y)ỵ<∞.
                
                The first of the above estimates gives us that Φ^ł is analytic in 0<|√(k^2+c_ł^2)|<√((c_ł+δ)^2-c_ł^2), which includes  a   δ-neighborhood of Σ,
                and the second estimate gives us analyticity of Φ^ in 
                |√(k^2+c_^2)|<√((c_ł+δ)^2-c_^2), which also includes  a δ-neighborhood of Σ.
                
                To prove part   (d) of the corollary  let us  
                introduce 
                
                
                
                
                
                 χ^±(k)=√(k^2+c^2_±), γ(k;c)=√(k-ı ck+ı c),   a^±_γ(k)=12γ(k,c_±)+1/γ(k,c_±)),b^±_γ(k)=12γ(k,c_±)-1/γ(k,c_±)),
                 andf_1(x;k):=∫_∞^x(L_1(x,y)a_γ(k)+L_2(x,y)b_γ(k))^ı(x-y)χ(k)ỵandf_2(x;k):= ∫_∞^x(-L_2(x,y)a_γ(k)+L_1(x,y)b_γ(k))^ı(x-y)χ(k)ỵwhere we drop everywhere the superscript±in the definition off_1(x;k)andf_2(x;k)andf_3^+(x;k):= ∫_+∞^x(-L_2^+(x,y)b_γ^+(k)+L_1^+(x,y)a_γ^+(k))^ı(y-x)χ^+(k)ỵ,f_4^+(x;k):=∫_+∞^x(L_1^+(x,y)b_γ^+(k)+L_2^+(x,y)a_γ^+(k))^ı(y-x)χ^+(k)ỵ.Then, substituting the expressionsΦ^-(x;k)=:[  φ^-(x;k) -ψ^-(x;k);  ψ^-(x;k)  φ^-(x;k) ]Φ^+(x;k)=:[ -ψ^+(x;k)  φ^+(x;k);  φ^+(x;k)  ψ^+(x;k) ],into the definition of the transition matrixT(k),we find the following expressions fora, b(which arex-independent, as they should be):
                
                    b(k) =   ^-ı x(χ^+(k)+χ^-(k)) [a^-_γ(k) +f_1^-(x;k)b_γ^+(k) + f^+_2(x;k)-.
                       .-b_γ^-(k) +f_2^-(x;k)(a_γ^+(k) + f_1^+(x;k)]
                
                and
                
                    a(k) =   ^ı (x(χ^+(k)-χ^-(k)) [(a_γ^-(k) +f_1^-(x;k)) (a_γ^+(k) +f_3^+(x;k))
                    -.
                       .-(b_γ^-(k) + f_2^-(x;k))(b_γ^+(k) +f_4^+(x;k) )
                    ].
                
                Integrating here by parts, and using estimates|L^±_j(x,y)|≤C(x) σ̂^±((x+y)/2),|∂_y L^±_j(x,y)|≤C(x) σ̂^±((x+y)/2),we see that all the integrals are convergent in the domaink∈ U_δ(Σ),and, furthermore, all the exponential terms with  dependence onkare estimated by^|x| |χ(k)|≤^C |x|,and hence are bounded ask→∞, | k|≤δ.The second property follows from the  integral representation in  Proposition <ref>.
                  The large k asymptotics in property 4  follows from integral representations in Proposition <ref> (b), and the possibility of integrating them by parts, as in Proposition <ref> (b).
                  
                
                § ZEROS OF A(K) IN IM√(K^2+C_Ł^2)≥0.
                
                In this section we 
                derive a RH problem in the case of higher order poles, which gives higher order solitons or breathers.
                
                
                
                Let k_0,Imk_0>0,k_0∉[ı c_ł,ı c_] be a zero of a(.) of the order n≥1, i.e.
                a(k_0)=⋯ a^(n-1)(k_0)=0,   a^(n)(k_0)≠0.
                Then there exist coefficients μ_0,⋯,μ_n-1
                independent on x,t, such that the following equalities hold:
                f_ł^(m)=∑_k=0^mC_m^kμ_m-kf_^(k),   m=0,⋯, n-1,   C_m^k:=m!/k!(m-k)!
                where f_±=[ φ_±,ψ_± ]^T,
                i.e.
                f_ł=μ_0f_,
                ḟ_ł=μ_0ḟ_+μ_1f_,
                f̈_ł=μ_0f̈_+2μ_1ḟ_+μ_2 f_,
                ⃛f_ł=μ_0⃛f_+3μ_1f̈_+3μ_2 ḟ_+μ_3 f_.
                ⋯
                 Denote f=Φ_ł,1,    g=Φ_,̊2
                the first and the second columns of Φ_ł,, respectfully.
                f,g are analytic in the upper half-plane.
                The spectral coefficient a(k) is given by the formula
                a(k)=[f(x,t;k),g(x,t;k)].
                
                The base of induction is the fact that if a(k_0)=0, then ∃μ_0  such that    f=μ_0 g.
                
                
                
                
                
                Further, suppose that the statement of the lemma is satisfied for m=1,⋯, m-1.
                By the Lebesgue formula,
                a^(m)(k_0)=∑_k=0^m[f^(m-k),g^(k)].
                In the above formula we split off the term containing f^m, and for all the smaller order derivatives of f we substitute their expression in terms of the derivatives of g. We get
                
                    a^(m)(k_0)=[f^(m),g]+∑_k=1^m∑_j=0^m-kC_m^kC_m-k^jμ_m-k-j[g^(j),g^(k)].
                
                Now, in the above expression each term [g^(j),g^(k)] with k≥1,k≠ j, will appear twice: once as
                C_m^kC_m-k^jμ_m-k-j[g^(j),g^(k)],
                and another time as 
                C_m^jC_m-j^kμ_m-j-k[g^(k),g^(j)].
                Since C_m^jC_m-j^k=C_m^kC_m-k^j=m!/j!k!(m-k-j)!,
                these two terms will cancel each other. The same is for j=k. Hence, the only terms which remain in (<ref>) are those corresponding to j=0, i.e.
                
                    a^(m)(k_0)=[f^(m),g]-∑_k=1^mC_m^kμ_m-k[g^(k),g]=
                    [f^(m)-∑_k=1^mC_m^kμ_m-kg^(k),g],
                
                and hence a^(m)(k_0)=0 implies the existence of a coefficient μ_m such that 
                f^(m)-∑_k=1^mC_m^kμ_m-kg^(k)=μ_mg,   or    f^(m)=∑_k=0^mC_m^kμ_m-kg^(k).
                This finishes the proof.
                
                
                
                
                
                 Let k_0,Imk_0>0,k_0∉[ı c_ł,ı c_] be a zero of a(.) of the n^th order, n≥1, i.e.
                a(k_0)=…=a^(n-1)(k_0)=0,   a^(n)(k_0)≠0;
                let the coefficients μ_0,…,μ_n-1  (independent of x,t!) be as in Lemma <ref>, i.e.
                
                    f_ł^(m)(k_0)=∑_j=0^mC_m^jμ_jf_^(m-j)(k_0),   m=0,…,n-1,      where C_m^j:=m!/j!(m-j)!
                
                are the binomial coefficients,
                and let T_1(x,t),…,T_n(x,t) be the coefficients in the Taylor expansion of the function 
                ^2ıθ(x,t;k)-2ıθ(x,t;k_0)/a(k) at the point k=k_0, i.e.
                
                    ^2ıθ(x,t;k)/a(k)=∑_q=1^nT_q(x,t) ^2ıθ(x,t; k_0)/(k-k_0)^q+𝒪(1),   k→ k_0.
                
                Then 
                
                    1/a(k)f_ł(x,t;k)^ıθ(x,t;k)-[∑_j=1^nA_j(x,t) ^2ıθ(x,t;k_0)/(k-k_0)^j]f_(x,t;k)^-ıθ(x,t;k)
                
                is regular in a vicinity of the point k=k_0 provided that 
                
                    A_j(x,t)=∑_m=j^nT_m(x,t)μ_m-j/(m-j)!=∑_m=0^n-jT_m+j(x,t)μ_m/m!,   j=1,…,n.
                
                 It is remarkable that the coefficients A_j are determined only by μ_j,j=0,…,n-1, and not by f_ and its derivatives at k=k_0.
                 Definition of A_j, j=1,…,n, requires up to the 2n-1^st derivative of a(.) at the point k=k_0, i.e. a^(n)(k_0),…,a^(2n-1)(k_0).
                 Regularity of the expression (<ref>) is equivalent to the regularity of 
                
                    1/a(k)f_ł(x,t;k)^2ıθ(x,t;k)-2ıθ(x,t;k_0)-[∑_j=1^nA_j(x,t)/(k-k_0)^j]f_(x,t;k).
                
                Let's first treat the left summand in (<ref>). We have f_ł=∑_m=0^n-11/m!f_ł^(m)(k_0)(k-k_0)^m+𝒪(k-k_0)^n,
                and substituting in the above formula the expressions (<ref>) of f_ł^(m) in terms of f_^(j), we obtain
                f_ł=∑_m=0^n-1∑_j=0^mμ_m-j f_^(j)(k_0)/j!(m-j)!(k-k_0)^m+𝒪(k-k_0)^n.
                Multiplying the above formula by (<ref>), we obtain
                
                    ^2ıθ(x,t;k)/a(k)^2ıθ(x,t;k_0)f_ł(x,t;k)=∑_p=1^n∑_m=0^n-pT_m+p(x,t)[∑_j=0^mμ_m-jf_^(j)(k_0)/j!(m-j)!](k-k_0)^-p+𝒪(1).
                
                The second summand in (<ref>) has the following decomposition in the Taylor series:
                
                    [∑_j=1^nA_j(x,t)/(k-k_0)^j]f_(x,t;k) = ∑_p=1^n∑_j=p^nA_j(x,t)f_^(j-p)(k_0)/(j-p)!(k-k_0)^-p+𝒪(1).
                
                Equating terms (k-k_0)^-p,p=1,…,n, in (<ref>) and (<ref>), we come to the system of equations for A_j:
                    ∑_j=p^nA_jf_^(j-p)(k_0)/(j-p)!=∑_m=0^n-pT_m+p∑_j=0^mμ_m-jf_^(j)(k_0)/j!(m-j)!,   p=1,…,n.
                
                The system (<ref>) allows   us to determine A_j, j=1,…,n, one by one, starting from A_n.
                However, it is not clear at this point that A_j do not depend on f_ and its derivatives. In order to solve the system (<ref>), let us rearrange the terms in (<ref>):
                
                    ∑_j=0^n-pA_j+pf_^(j)(k_0)/j!=∑_j=0^n-p[∑_m=j^n-pT_m+pμ_m-j/(m-j)!]f_^(j)(k_0)/j!,   p=1,…,n.
                
                The above expression (<ref>) is satisfied provided that 
                
                    A_j+p=∑_m=j^n-pT_m+pμ_m-j/(m-j)!,   p=1,…,n,   j=0,…,n-p,
                
                however, (<ref>) is not yet the definition of A_j+p, since the r.-h.-s. might depend on different choices for j,p with j+p. However, reorganizing terms in (<ref>), we obtain
                A_j(x,t)=∑_m=j^nT_m(x,t)μ_m-j/(m-j)!,   j=1,…,n.
                
                 §.§ Handling pole conditions
                
                The analysis of  RH problems with  simple poles  was first introduced  in <cit.>  for KdV  (see also <cit.> for the CH equation).
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                
                Here we explain how to proceed, if our spectral problem has poles of multiple order.
                First of all, the pole condition 3 of the RH problem <ref> will change to the following conditions:
                forκ_j, κ_j>0, κ_j>0,letM_1, M_2be  the first and the second columns ofM.Then the following pole conditions hold:
                
                    M_1(k) - [∑_l=1^n_jν_j^(l) e^2ıθ(x,t;k)/(k-κ_j)^l]M_2(k) = 𝒪(1) , k→κ_j,
                       
                    M_1(k) - [∑_l=1^n_jν_j^(l) e^2ıθ(x,t;k)/(-1)^l(k+κ_j)^l]M_2(k) = 𝒪(1) , k→ -κ_j,
                    
                       
                    M_2(k) + [∑_l=1^n_jν_j^(l) e^-2ıθ(x,t;k)/(k-κ_j)^l]M_1(k) = 𝒪(1) , k→κ_j,
                       
                    M_2(k) + [∑_l=1^n_jν_j^(l) e^-2ıθ(x,t;k)/(-1)^l(k+κ_j)^l]M_1(k) = 𝒪(1) , k→ -κ_j,
                    
                       
                    M_2(k)=𝒪(1), k→κ_j,  k → -κ_j,      M_1(k)=𝒪(1), k→κ_j k→ -κ_j.
                
                
                  §.§.§ Applications: Solitons and breathers of multiple order
                
                They are generated by the following meromorphic RH problem.
                
                Find a meromorphic 2×2 matrix-valued function M(x,t;k) such that
                
                  * M is meromorphic in ℂ with poles at κ,κ, -κ,-κ,
                for some  κ≥0,κ>0;
                  *  pole conditions, upper half-plane:
                M_1(k)-[∑_j=1^nA_j ^2ıθ(κ)/(k-κ)^j]M_2(k)=𝒪(1),   k→κ,M_1(k)-[∑_j=1^n(-1)^j A_j ^2ıθ(-κ)/k+κ^j]M_2(k)=𝒪(1),   k→ -κ,
                lower half-plane:
                M_2(k)-[∑_j=1^n-A_j ^-2ıθ(κ)/k-κ^j]M_1(k)=𝒪(1),   k→κ,M_2(k)-[∑_j=1^n(-1)^j-1A_j  ^2ıθ(κ)/(k+κ)^j]M_1(k)=𝒪(1),   k→ -κ;
                  *  asymptotics: M(x,t;k)→ I as k→∞.
                HereA_j = A_j(x,t)are as in (<ref>), (<ref>), i.e.
                
                  *  simple pole n=1, a(κ)=0, ȧ(κ)≠0. In this case we have 
                T_1=^2ıθ(κ)/ȧ(κ),   A_1=μ_0T_1 .
                Here θ(k)=4k^3t+xk, and the dependence on x,t comes only from θ(k).
                  *  double-pole n=2,  a(κ)=0, ȧ(κ)=0, ä(κ)≠0. In this case we have 
                T_2=2^2ıθ(κ)/ä(κ),   T_1=2ı^2ıθ(κ)[6ä(κ)x+12κ^2t+ı⃛a(κ)]/3ä(κ)^2,
                and A_2=T_2μ_0,   A_1=μ_0T_1+μ_1T_2,
                  *  triple-pole n=3,  a(κ)=0, ȧ(κ)=0, ä(κ)=0, ⃛a(κ)≠0. In this case we have 
                T_3=6^2ıθ(κ)/⃛a(κ),   T_2=24 ı ⃛a(κ)x+12κ^2t-3a^(4)(κ)/2⃛a(κ)^2^2ıθ(κ),
                    T_1^-2ıθ(κ)=-12[(x+12κ^2t)^2-12ıκ t]/⃛a(κ)-
                    
                    -3ı a^(4)(κ)x+12κ^2t/⃛a(κ)^2
                    +15a^(4)(κ)^2 -12a^(3)(κ)a^(5)(κ)/40⃛a(κ)^3,
                 A_3=T_3μ_0,   A_2=μ_0T_2+μ_1T_3,   A_1=μ_0T_1+μ_1T_2+μ_2/2T_3.
                
                Taking into account symmetry (<ref>), the solution can be found in the formM(k)=[   1+∑_j=1^nα_j(x,t)/(k-κ)^j + ∑_j=1^n(-1)^jα_j(x,t)/(k+κ)^j ∑_j=1^n-β_j(x,t)/(k-κ)^j + ∑_j=1^n(-1)^j-1 β_j(x,t)/(k+κ)^j;     ∑_j=1^nβ_j(x,t)/(k-κ)^j + ∑_j=1^n(-1)^jβ_j(x,t)/(k+κ)^j 1+ ∑_j=1^nα_j(x,t)/(k-κ)^j + ∑_j=1^n(-1)^j α_j(x,t)/(k+κ)^j ],(Ifκ=-κ≠0,then we make a straightforward reduction of the above expression).
                Then we find the solution of the MKdV by the formula
                
                    q(x,t)=2ıβ_1(x,t)-β_1(x,t)=-4β_1(x,t).
                
                
                
                
                Ifκ=0, κ>0, then we have a multiple soliton, ifκ>0, κ>0,then we have a multiple breather.
                As was pointed out in <cit.>, double-pole soliton can be obtained as a limit of a breather withκ→0.1cm
                 Acknowledgment.
                This manuscript was partially developed while T.G. was Chair Morlet at the  Centre International de Rencontres Mathematiques, (CIRM), Luminy France.
                T.G. thanks CIRM for  the warm hospitality.  T.G. wish to thank   GNFM (Gruppo Nazionale di Fisica matematica) and INDAM.
                
                
                
                
                
                
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