Solitary wave solutions of a Whitham-Boussinesq system
Evgueni Dinvay Dag Nilsson





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    Maxime Portaz

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m.portaz@qwant.com





Hicham Randrianarivo

Qwant Research

h.randrianarivo@qwant.com
Adrien Nivaggioli

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a.nivaggioli@qwant.com
Estelle Maudet

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e.maudet@qwant.com
Christophe Servan

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c.servan@qwant.com
Sylvain Peyronnet

Qwant Research

s.peyronnet@qwant.com

    December 30, 2023
=======================================================================================================================================================================================================================================================================================================================================================








^1Department of Mathematics,
University of Bergen,


Postbox 7800, 5020 Bergen, Norway.


^2Department of Mathematical Sciences, NTNU,


NO-7491 Trondheim, Norway.


E-mail addresses:
Evgueni.Dinvay@uib.no;
Dag.Nilsson@ntnu.no










Current addresses:


^1Inria Rennes - Bretagne Atlantique,


Campus universitaire de Beaulieu Avenue du Général Leclerc,


35042 Rennes Cedex,
France.


^2Department of Mathematics, Saarland University,


Campus E24, 66123 Saarbrücken, Germany.


E-mail addresses:
Evgueni.Dinvay@inria.fr;
nilsson@math.uni-sb.de









The travelling wave problem
for a particular bidirectional Whitham system
modelling surface water waves is under consideration.
This system firstly appeared in <cit.>,
where it was numerically shown to be stable and a good approximation
to the incompressible Euler equations.
In subsequent papers <cit.>
the initial-value problem was studied and
well-posedness in classical Sobolev spaces was proved.

Here we prove existence of solitary wave solutions
and provide their asymptotic description.
Our proof relies on a variational approach and
a concentration-compactness argument.
The main difficulties stem from the fact
that in the considered Euler-Lagrange equation
we have a non-local operator of positive order
appearing both in the linear and non-linear parts.

Our approach allows us to obtain solitary waves
for a particular Boussinesq system as well.







§ INTRODUCTION






 §.§ Motivation and background



In this work we consider solitary wave solutions of the Whitham-Boussinesq system

    K η_t + K v_x + (η v)_x
    	   = 0
    	,
    
    	K v_t + η_x + vv_x
    	   = 0
    	,


where 
K is a Fourier multiplier operator meaning
(Kf)(ξ) = k(ξ) f(ξ)
for any tempered distribution f ∈𝒮'(ℝ).
Here ℱ stands for the Fourier transform

    ℱ(f)(ξ)
    	=
    	f(ξ)
    	=
    	∫_ℝf(x)e^-ixξ dx
    	.


A solitary wave is a solution of the form

    η(x,t) = η(x+ct)
    	,	v(x,t)=v(x+ct),

with η(x+ct),  v(x+ct)→ 0,
as x+ct→∞.




How about moving Definition 1.1 here? I think this would make the introduction a bit more clear. Also, instead of using K in the above system we could use L directly
I prefer to leave it like this
unless the next referees recommend us to do it...
Sure, we can keep it like this for now.




The Fourier multipliers we will be considering in this paper include


    K = D/tanh(D)


with D=-i∂_x, and symbol
k(ξ) = ξ / tanhξ
	.
Note that this operator is of order one, it is equivalent to the Bessel potential J =(1-∂_x^2)^1/2
associated with the symbol
⟨ξ⟩
	=
	√(1 + ξ^2)
,
 since
ξ / tanhξ≃⟨ξ⟩
	.

Such choice of K is motivated by the water wave problem,
when η denotes the surface elevation
and v is the fluid velocity at the surface. 
Another example is the operator K = 1 - b ∂_x^2, with the corresponding symbol k(ξ) = 1 + b ξ^2.  For this choice of K, (<ref>)–(<ref>) becomes a (-b,b,0,b)-Boussinesq system. We recall here that the general (a, b, c, d)-Boussinesq system is of the form


    η_t + v_x + (η v)_x
    		+ av_xxx - bη_xxt   = 0
    	,
    
    		v_t + η_x + vv_x
    		+ cη_xxx - dv_xxt   = 0
    	,


and was derived in <cit.>.
It was shown in <cit.>
that (<ref>) exhibits
travelling waves for a, c < 0 and b = d.
The main result of the present paper (Theorem <ref>) implies that (<ref>)
has solitary waves in the case
a = -b, c = 0, d = b
for any b > 0.
To our knowledge this is a new result.





Note that with b = 1/3 this choice corresponds
to a physically relevant case
<cit.>.


The model (<ref>)-(<ref>)
with K defined by (<ref>)
was formally derived in <cit.>
from the incompressible Euler equations to model
fully dispersive shallow water waves.
In fact it can be regarded as a fully dispersive
improvement of the (-1/3, 1/3, 0, 1/3)-Boussinesq system.




System (<ref>)–(<ref>) was introduced
in <cit.> as an extension
of the unidirectional Whitham equation


    ∂_t η
    	+ √(K^-1)∂_x η
    	+ 3/2η∂_x η = 0


allowing two-way wave propagation.
Equation (<ref>) was proved to be locally well-posed
in Sobolev spaces H^s with s > 3/2 in <cit.>.
In recent years, several interesting phenomena predicted by Whitham
has been confirmed, for example, a solitary wave regime close to KdV <cit.>,
the existence of a wave of greatest height
<cit.>,
the existence of shocks <cit.>,
and modulational instability of 
steady periodic waves <cit.>.


Apart from (<ref>)-(<ref>)
in recent years several 
bidirectional extensions of (<ref>) have been put forward to,
as for example

    η_t
    	   =
    	- K^-1v_x-(η v)_x,
    
     v_t   =-η_x-vv_x


and


    η_t
    	   =
    	-v_x-(η v)_x,
    
    	v_t
    	   =-K^-1η_x-vv_x
    	.



The first system (<ref>) has a Hamiltonian structure
and was formally derived in <cit.>
from
the incompressible Euler equations to model fully dispersive
shallow water waves whose propagation is allowed to be both left- and rightward, and appeared in
<cit.>
as a full dispersion system in the Boussinesq regime with the dispersion of the water
waves system.
There have been several investigations on this system:
local well-posedness <cit.>
(in homogeneous Sobolev spaces at a positive background), a
logarithmically cusped wave of greatest height
<cit.>.
In <cit.>
they
impose an additional non-physical condition
η⩾ C > 0.
Kalisch and Pilod <cit.>
have proved local well posedness
for a surface tension regularisation of System
(<ref>)
with ( 1 - β∂_x^2 ) K^-1 standing instead
of K^-1 in the first equation.
So they managed to
remove the positivity assumption η > 0.
However, the maximal time of existence for their
regularisation is bounded by the capillary parameter β > 0.
The existence of solitary waves for this system was established in 
<cit.>.


The second system (<ref>) was introduced in <cit.>
in order to better model modulational instabilities.
Indeed, it was found in <cit.> that,
when including the effects of surface tension,
the system (<ref>) gives accurate predicitions of
Benjamin-Feir instabilities.

It was pointed out in <cit.> that System (<ref>)
has also a Hamiltonian structure.
To the authors knowledge
neither well-posedness nor the existence of solitary waves
for the system (<ref>) have been established yet.


There are also numerical results on
the validity of the both systems (<ref>), (<ref>)
for modelling waves on shallow water <cit.>,
numerical bifurcation and spectral stability <cit.>. 


System (<ref>)–(<ref>)
has been recently shown to be well-posed in
H^s(ℝ) × H^s+1/2(ℝ)
with s > -1/10
in <cit.>.
Moreover, the result is global
for s ⩾ 0
if the initial data (η_0, v_0)
has sufficiently small
L^2 × H^1/2-norm.
The latter is the main advantage of 
Equations (<ref>)–(<ref>) comparing
with the other models (<ref>), (<ref>).













There is a fully-dispersive Green-Naghdi type model introduced by Duchêne, Israwi and Talhouk
<cit.> that was not considered in <cit.>.
Existence of solitary wave solutions for this system
was established in <cit.>.
For more discussion on the Cauchy problem
and rigorous justification of the various Whitham related equations
we refer to <cit.>.




 
The main aim of the current paper is to prove the existence
of solitary wave solutions for (<ref>)-(<ref>)
with K being an admissible Fourier multiplier,
see Definition <ref> below. Both K given by (<ref>) and K = 1 - b ∂_x^2 are examples of admissible Fourier multipliers.
Note that the existence of solitary waves supports
validity of System (<ref>)-(<ref>)
from physical perspective
as a weakly nonlinear wave model.
We use a variational approach together with Lion's method of
concentration-compactness <cit.>
to establish the existence of
solitary wave solutions of (<ref>)–(<ref>).













This approach has been used extensively to prove
existence of solitary wave solutions to equations of the form 

    u_t+Lu_x+n(u)_x=0,

where L is a Fourier multiplier operator of order s,
essentially meaning that the symbol of the operator can be bounded from above and below
by ξ^s up to a constant,
and n(u) is a homogeneous nonlinear term.
Under the travelling wave ansatz u = u(x + ct),
equation (<ref>) becomes

    cu+Lu+n(u)=0.

In <cit.> existence and stability of solitary wave solutions for long wave model equations of the form (<ref>), with s≥ 1, was established. This approach was later used in <cit.> to prove existence of solitary waves for an equation used to model stratified fluids, with s=1,  and was later generalized in <cit.> to s≥ 1.
A class of Whitham type equations of the form (<ref>) was studied in <cit.>,
with a Fourier multiplier operator of negative order.
In this case the resulting functional in the constrained minimization problem is not coercive. This makes the application of the concentration compactness theorem a lot more technical, requiring the authors to use a strategy developed in <cit.>
and first consider a related penalized functional acting on periodic functions.
In the recent work <cit.> an entirely different
approach to proving the existence of solitary wave solutions
of the Whitham equation, based on the implicit function theorem
instead, was presented.
Arnesen proved existence of solitary wave solutions to
two different classes of model equations <cit.>,
one of them of the form (<ref>), for s>0.
Results, similar and previous to those of Arnesen,
were obtained in <cit.>
in application to two particular cases, namely,
the fractional Korteweg-de Vries
and the fractional Benjamin-Bona-Mahony equations.
The case when the nonlinearity n is allowed to be inhomogeneous was considered in <cit.>,
where the author proved the existence of solitary wave solutions of (<ref>), for operators of positive order and with weak assumptions on the regularity of the symbol.  



These methods have also been applied to bidirectional
Whitham type equations.
As mentioned above, in <cit.> the authors established the existence of solitary waves for the class of modified Green–Naghdi equations introduced in
<cit.>,
and in <cit.>
the authors proved the existence of solitary waves
for (<ref>).
Just as in <cit.>,
both of the functionals appearing in
<cit.>
are noncoercive, so the minimization arguments adapted to noncoercive functionals developed in <cit.>
are used in order to obtain the existence of minimizers. In addition, the Fourier multiplier operator is entangled with the nonlinearity in <cit.>, which makes the proofs more technical.





 §.§ The minimization problem

 



We formulate the problem in the variational settings.
A Hamiltonian structure <cit.>
of System (<ref>)–(<ref>)
allows us to do this in a relatively straightforward way.
Indeed, under the travelling wave ansatz (<ref>),
equations (<ref>)–(<ref>) can be written as

    Kv+η v+cKη   =0,
    η+v^2/2+cKv   =0,

where the constants of integration are set to zero since we are considering solitary wave solutions.



Regarding the Hamiltonian and momentum

    ℋ(η,v)   =1/2∫_ℝη^2+vKv+η v^2 dx,
    ℐ(η,v)   =∫_ℝη Kv dx,

one can notice that Equation
(<ref>) can be written as

    d_vℋ+cd_vℐ=0,

and Equation (<ref>) as 

    d_ηℋ+cd_ηℐ=0.



One can try to proceed further with this formulation
as was done for the
(a, b, c, d)-Boussinesq system (<ref>)
in <cit.>.
However, instead of looking for critical points
of the functional
ℋ + c ℐ
we reduce System
(<ref>)-(<ref>) to

a single travelling wave equation
that can in turn be interpreted as a constrained minimization problem.
Note that our approach allows us to extend the results
obtained in <cit.>.




I think this need to be elaborated on. Are you here referring to the asymptotic description of our solitary waves? Overall it would be good if we could motivate our approach a bit more. For example, if we include strong surface tension (greater than 1/3), then we could probably get a stability result if we would have used the approach <cit.>. However, since we have surface tension zero, there is no hope for us to get stability, so at least there does not seem to be a benefit to using the approach in <cit.>. When we first started this work I thought that the approach in <cit.> could not be used in our case, but it may actually be possible. I will try to think about this some more so that we can write something more concrete here.


In both works by Chen ( <cit.>
and with the large surface tension )
they impose restrictions:
d = b, a < 0 and c < 0.
We have regarded some kind of limiting case:
c = 0!
I meant that we have extended the results to the limiting case
(that was not considered by Chen)
and that
is all I can say for sure........
Our system corresponds in some sense to c = 0,
and so I believe the Chen approach will not work at some point....
If we add the surface tension ϰ > 0 to our system then
c = - ϰ < 0 in some sense, and so I believe
in Chen's approach here .......
I actually think that extension of the Chen approach
will be too straightforward and we might have difficulty
because of this with publishing later.
However, this concerns only the surface tension case.
In the current project most likely Chen's approach is useless
because of c = 0.
Okay, this is a good observation. I guess if the referee should remark about this we can add an explanation




We can derive a travelling wave equation in the following way. 
In (<ref>)–(<ref>) we make the change of variable v=K^-1/2ṽ, which yields the new system

    K^1/2ṽ+η(K^-1/2ṽ)+cKη   =0,
    η+(K^-1/2ṽ)^2/2+cK^1/2ṽ   =0
    
    .

From (<ref>) we get that

    η=-(K^-1/2ṽ)^2/2-cK^1/2ṽ,

and inserting this into (<ref>) yields

    ṽ-K^-1/2((K^-1/2ṽ)^3/2)-cK^-1/2((K^1/2ṽ)(K^-1/2ṽ))-cK^1/2((K^-1/2ṽ)^2/2)-c^2Kṽ=0.

Here we make the change of variables ṽ=cu so that (<ref>) becomes


    1/c^2u-K^-1/2((K^-1/2u)^3/2)-K^-1/2((K^1/2u)(K^-1/2u))-K^1/2((K^-1/2u)^2/2)-Ku=0.


Now let us show that Equation (<ref>) represents
an Euler-Lagrange equation for some functional.
Indeed, regard the surface elevation and velocity
defined by u as follows


    η_u
    	   =
    	-c^2( (K^-1/2u)^2/2 + K^1/2u )
    	, 
    
    	v_u
    	   =
    	cK^-1/2u
    	,


and note that

    ℋ(η_u,v_u)+cℐ(η_u,v_u)=c^4[-1/2∫_ℝ uKu+(K^1/2u)(K^-1/2u)^2+(K^-1/2u)^4/4 dx+1/2c^2∫_ℝ u^2 dx]
    ,

which leads us to define

    ℰ(u)   =1/2∫_ℝuKu+(K^1/2u)(K^-1/2u)^2+(K^-1/2u)^4/4 dx,
       =1/2∫_ℝ(K^1/2u + 1/2 (K^-1/2u)^2)^2
    	 dx
    𝒬(u)   =1/2∫_ℝu^2 dx.

We then note that equation (<ref>) can be written as

    dℰ(u)+λd𝒬(u) = 0,

where λ=-1/c^2.
Hence, in order to find solutions of (<ref>) we can consider the constrained minimization problem


    inf _u ∈ U_qℰ(u)
    	    
    	U_q =
    	{
    		u ∈ H^1/2(ℝ)  𝒬(u) = q
    	}
    	.




Instead of working with the specific Fourier multiplier K,
we will work with a more general class of Fourier multipliers,
and thus a more general constrained minimization problem.
The proof does not get much more complicated
if we consider a general class of multipliers.
Moreover, as was mentioned in the introduction,
it allows us to treat also
a Boussinesq system not considered from this perspective before.








Let operator L be a Fourier multiplier,
with symbol m, i.e.

    ℱ(Lf)(ξ) = m(ξ) f(ξ).

We say that L is admissible
if m is even, m(0)>0 and for some s'>1 and s>1/2
the symbol satisfies the following restrictions.






  * The function ξ↦m(ξ)/⟨ξ⟩^s
is uniformly continuous, and

    0<m(ξ)-m(0)    ≲ξ^s'  ξ⩽ 1
    ,
    
    
    
    
    m(ξ)-m(0)    ≃ξ^s
      ξ>1
    .




  * For each ε> 0 the kernel
of operator L^-1/2 satisfies

    ℱ^-1(m^-1/2)
    	∈ L^2(ℝ∖ (- ε, ε)).

There exists
p ∈ (1, 2) ∩ [2 / (s+1),2)




such that

    ℱ^-1(m^-1/2)
    	∈ L^p(- 1, 1)
    	.










The symbol m(ξ)=ξ/tanh(ξ) satisfies the conditions
of Definition <ref> with s = 1 and s'= 2
as was shown in
<cit.>.
For the symbol m(ξ)= 1 + b ξ^2 with b > 0
we have s = s' = 2 and
m^-1/2∈ L^2(ℝ)
	.
In particular,
ℱ^-1(m^-1/2)
	∈ L^2(ℝ)
and so (<ref>), (<ref>) hold.


We have the corresponding functional


    ℰ(u)=1/2∫_ℝ(L^1/2u + 1/2 (L^-1/2u)^2)^2
    	 dx






defined on H^s/2(ℝ).
Our main goal is then to obtain a solution of the minimization problem

    inf _u ∈ U_qℰ(u)
    	    
    	U_q =
    	{
    		u ∈ H^s/2(ℝ)  𝒬(u) = q
    	}
    	.


For convenience we separate ℰ into the functionals

    ℒ(u)   =1/2∫_ℝuLu dx,
    𝒩_c(u)   =1/2∫_ℝL^1/2u(L^-1/2u)^2 dx,
    𝒩_r(u)   =1/2∫_ℝ(L^-1/2u)^4/4 dx

so that

    ℰ(u) = ℒ(u) + 𝒩(u)

where

    𝒩(u) = 𝒩_c(u) + 𝒩_r(u)
    	.




We are now ready to state our main results. 



Let D_q be the set of minimizers of ℰ over U_q. There exists q_0>0 such that for each q∈ (0,q_0), the set D_q is nonempty and u_H^s/2^2≲ q for u∈ D_q.
Each element of D_q is a solution of the Euler–Lagrange equation

    λ u+L^-1/2((L^-1/2u)^3/2)+L^-1/2((L^1/2u)(L^-1/2u))+L^1/2((L^-1/2u)^2/2)+Lu=0.

The Lagrange multiplier λ satisfies 

    m(0)/2< -λ< m(0)-Dq^β,

where β=s'/2s'-1 and D is a positive constant.

Here and throughout the paper we write f≲ g, when f/g is uniformly bounded from above, and f≃ g when f≲ g≲ f. Our other main result concerns the asymptotic behavior of  travelling wave solutions of (<ref>)–(<ref>).



If L=K defined by (<ref>)
then there exists q_0>0 such that for any
q∈(0,q_0) each minimizer u∈ D_q belongs to H^r(ℝ)
for any r ⩾ 0 with u_H^r^2≲ q,
and moreover, it satisfies the following long wave asymptotics


    sup_u∈ D_qinf_x_0∈ℝ
    	q^-2/3u(q^-1/3·)-ψ_KdV(·-x_0)
    	_H^1(ℝ)≲ q^1/6
    ,


whereas the corresponding surface elevation (<ref>)
and speed (<ref>) satisfy


    sup_u∈ D_qinf_x_0∈ℝ
    	q^-2/3η_u(q^-1/3·)
    	+
    	ψ_KdV(·-x_0)
    _H^1/2(ℝ)≲ q^1/6
    ,
    sup_u∈ D_qinf_x_0∈ℝ
    	q^-2/3 v_u(q^-1/3·)
    	+
    	ψ_KdV(·-x_0)
    _H^3/2(ℝ)≲ q^1/6
    ,

where 

    ψ_KdV(x)=-λ_0^2(1/2√(3λ_0)x)

and λ_0=3/√(16). In addition, the Lagrange multiplier λ satisfies

    λ=-1+λ_0q^2/3+𝒪(q^5/6).


 We discuss here briefly how to prove
Theorems <ref>, <ref>. 

The main ingredient in proving Theorem <ref> is Lion's concentration compactness theorem <cit.>:


Any sequence {e_n}_n∈ℕ⊂ L^1(ℝ) of non-negative functions such that

    lim_n→∞∫_ℝe_n d x= I>0

admits a subsequence, denoted again {e_n}_n∈ℕ, for which one of the following phenomena occurs.


  * (Vanishing) For each r>0, one has

    lim_n→∞( sup_x∈ℝ∫_x-r^x+re_n d x)=0.


  * (Dichotomy) There are real sequences {x_n}_n∈ℕ,{M_n}_n∈ℕ,{N_n}_n∈ℕ⊂ℝ and I^* ∈(0,I) such that M_n,N_n→∞, M_n/N_n→ 0, and

    ∫_x_n-M_n^x_n+M_ne_n d x→ I^*    and   ∫_x_n-N_n^x_n+N_ne_n d x→ I^*,

as n→∞.

  * (Concentration) There exists a sequence {x_n}_n∈ℕ⊂ℝ with the property that for each ϵ>0, there exists r>0 with 

    ∫_x_n-r^x_n+r e_n 𝕕 x ≥ I-ϵ,

for all n∈ℕ.


We will apply this theorem to e_n=u_n^2, where {u_n}_n=1^∞ is a minimizing sequence, and show that the vanishing and dichotomy scenarios cannot occur.
Then we obtain a convergent subsequence of {u_n}_n=1^∞
using the concentration scenario.
The functional ℰ is similar to the corresponding functionals appearing in <cit.>, in the sense that the Fourier multiplier and the nonlinerity are entangled.
However, in contrast with <cit.>,
our functional ℰ is 
bounded from below,

hence the penalization argument
of  <cit.>
is not necessary in our case. 


In <cit.> the exclusion of dichotomy
gets more technical due to the entanglement of
the Fourier multiplier with the nonlinearity,
and this is true for the present work as well.
In contrast, the exclusion of the vanishing scenario is straightforward
in <cit.>,
while this is not the case in the present work.
This is due to the fact that in <cit.>
the constrained minimization problem is formulated in H^s(ℝ),
s> 1/2, allowing the use of the embedding
H^s(ℝ)↪ L^∞(ℝ),
while our problem is formulated in H^s/2(ℝ),
preventing the use of this embedding.
Instead we show that if {u_n}_n=1^∞ is vanishing,
then L^-1/2u_n is vanishing as well,
which leads to a contradiction.
In order to show that L^-1/2u_n is vanishing we make use
of the integrability assumptions (<ref>), (<ref>)
imposed on the kernel of L,
and this is the only instance where these assumptions are used.
Our other assumptions on L as are similar to those in <cit.>,
and we are able to adopt many of the methods used in that paper
to our present work.
We give here a brief explanation on where the remaining assumptions
on m in Definition <ref> are used.
The uniform continuity of m(ξ)/⟨ξ⟩^s
is used to prove Lemma <ref>,
which in turn is used to exclude the dichotomy scenario.
The upper bound in (<ref>) is used to prove
Proposition <ref>,
which allows us to define near minimizers.
In Proposition <ref> we also make use of the lower bound
on s', which tells us that the number β = s'/(2s'-1)
is strictly less than 1.
The latter and
the lower bound in (<ref>) are used to prove subadditivity
of I_q in Proposition <ref>.
The lower bound in (<ref>) essentially gives us coercivity
and enables us to obtain a uniform upper bound on near minimizers
in H^s/2-norm, in Proposition <ref>.
The lower bound on s is needed when excluding
the vanishing scenario in Proposition <ref>.
Here we make use of the fact that L^-1/2u ∈ H^s,
for u∈ H^s/2, and since s>1/2 we can conclude
that L^-1/2u ∈ L^∞.
Finally, the upper bound in (<ref>) is essentially used to estimate
L^1/2u_n_L^2≲u_n_H^s/2,
where u_n is a minimizing sequence,
and we know from before that u_n is uniformly bounded
in H^s/2-norm.
Hence L^1/2u_n_L^2 is also uniformly bounded,
and this fact is used in Proposition <ref> to exclude dichotomy,
and also in Proposition <ref> while
proving existence of a minimizer.


Theorem <ref> is established using more standard arguments,
see for example <cit.>.





§ TECHNICAL RESULTS




The current section is devoted to the general properties
of the functionals introduced above.
We start with a useful proposition on continuity
of symbol m(ξ) described by Definition <ref>.

There is a function ω : ℝ→ [0,∞), bounded from above by a polynomial, with lim_λ→ 0ω(λ)=0, such that

    m(ξ)-m(η)≤ω(ξ-η)⟨ξ⟩^s/2⟨η⟩^s/2.
 



The proof is given in <cit.>.


The following functional estimates will be used a lot
in the text below,
sometimes without references.


For any u∈ H^s/2(ℝ) one has

    ℒ(u)≃u_H^s/2^2.



This is immediate from Definition <ref>.


For any s>1/2 and u∈ H^s/2(ℝ) one has

    𝒩_c(u)   ≲u_L^2^2u_H^s/2,
    𝒩_r(u)   ≲u_L^2^4.



Inequality (<ref>) follows from the Sobolev embedding

    𝒩_r(u)=1/8L^-1/2u_L^4^4≲∂_x^1/4L^-1/2u_L^2^4≲J^1/4-s/2u_L^2^4≲u_L^2^4.

Inequality (<ref>) follows from (<ref>) and Hölder's inequality.


For s>1/2 and u,h∈ H^s/2(ℝ)
the Fréchet derivative of ℰ satisfies

    dℰ(u)(h)≲u_H^s/2(1+u_L^2+u_L^2^2)h_H^s/2



We first note that

    dℒ(u)(h)≲u_H^s/2h_H^s/2.

Next consider

    d𝒩_c(u)(h)=1/2∫_ℝL^1/2h(L^-1/2u)^2+2uL^1/2(L^-1/2uL^-1/2h) dx,

where

    L^1/2h(L^-1/2u)^2_L^1≤L^1/2h_L^2L^-1/2u_L^4^2≲u_L^2^2h_H^s/2,


    uL^1/2(L^-1/2uL^-1/2h)_L^1   ≤u_L^2L^1/2(L^-1/2uL^-1/2h)_L^2
       ≲u_L^2L^-1/2uL^-1/2h_H^s/2
       ≲u_L^2L^-1/2u_H^s/2L^-1/2h_H^s
       ≲u_L^2^2h_H^s/2.

Using the above estimates in (<ref>), we immediately get that

    d𝒩_c(u)(h)≲u_L^2^2h_H^s/2.

In a similar way we find that

    d𝒩_r(u)(h)≲u_L^2^3h_H^s/2,

which concludes the proof.

We next record a decomposition result for 𝒩_c.

Let u∈ H^s/2(ℝ).
Then

    𝒩_c(u)=1/2√(m(0))∫_ℝu^3 dx+𝒩_c1(u)+𝒩_c2(u)+𝒩_c3(u),

where 

    𝒩_1c(u)   =√(m(0))/2∫_ℝu((L^-1/2-m^-1/2(0))u)^2 dx
    𝒩_2c(u)   =∫_ℝu^2(L^-1/2-m^-1/2(0))u dx
    𝒩_3c(u)   =1/2∫_ℝ(L^-1/2u)^2(L^1/2-m^1/2(0))u dx,

and

    𝒩_2c(u)   ≤u_L^4^2(L^-1/2-m^-1/2(0))u_L^2
    ,
    𝒩_3c(u)   ≲u_L^2^2(L^1/2-m^1/2(0))u_L^2.



The proof is straightforward and is therefore omitted.


Before we continue we want to make a remark on the convolution theorem.
According to our choice of the Fourier transform normalisation,
for any two functions f and g we have

    ℱ (fg)
    	= 
    	1/2πf * g

where star stands for convolution.



	The functional ℰ defined by (<ref>)
	is translation invariant.
	In other words,
	for any u ∈ H^s/2(ℝ)
	then
	ℰ(u_h) = ℰ(u)
		,
where u_h(x) = u(x - h) denotes translation by h ∈ℝ.





Due to the property u_h(ξ) = e^-ihξu(ξ)
and the Plancherel theorem we have


    ℰ(u_h) =
    	1/4π∫_ℝ|
    		√(m(ξ))u_h(ξ)
    		+ 1/2ℱ(( L^-1/2u_h )^2) (ξ)
    	|^2
    	 dξ
    
    	=	
    	1/4π∫_ℝ|
    		e^-ihξ√(m(ξ))u(ξ)
    		+ 1/2 e^-ihξℱ(( L^-1/2u )^2) (ξ)
    	|^2
    	 dξ
    	= ℰ(u)


where we have also used the fact that the Fourier transform
of multiplication is convolution
of Fourier transforms
up to a normalization constant.



































In the following lemma we provide a slightly sharper
estimate for 𝒩_c.
It will be the first step
towards the non-vanishing proof given below.



	For s > 1/2 the following estimate hold true	
	
    | 𝒩_c(u) | ≲‖ u ‖_L^2 (ℝ)^2
    		‖ L^-1/2u ‖_L^∞ (ℝ)





	Clearly, L^-1/2u ∈ L^∞ (ℝ)
	and so applying a Kato–Ponce type estimate obtain
	
    | 𝒩_c(u) |
    		≲‖ u ‖_L^2‖ L^1/2( L^-1/2u )^2 ‖_L^2≲‖ u ‖_L^2‖ J^s/2( L^-1/2u )^2 ‖_L^2
    ≲‖ u ‖_L^2‖ J^s/2 L^-1/2u ‖_L^2‖ L^-1/2u ‖_L^∞≲‖ u ‖_L^2^2
    		‖ L^-1/2u ‖_L^∞
    		.




We finish this section with a lemma which will be used when ruling out the dichotomy scenario.



Let φ∈𝒮(ℝ), and let A_r H^s/2(ℝ)→ H^s/2(ℝ), B_r L^2(ℝ)→ L^2(ℝ) be the operators

    A_rf   =[L,φ(./r)]f,
    
    B_rf   =[L^-1/2,φ(./r)]f.

Then the operator norms

    A_r
    	,
    	B_r
    	→ 0
    	  r →∞.



We follow the proof of <cit.>.
Let φ_r(x)=φ(x/r).
Using Lemma <ref>, we find that for f,g∈ H^s/2(ℝ)

    ⟨ A_rf,g⟩   =
    1/2π|
    	∫_ℝ∫_ℝφ_r(η)
    	f(ξ-η)(m(ξ)-m(ξ-η))g(ξ) dη dξ|
    
       ≲∫_ℝ|
    φ_r(η)ω(η)
    |
    ∫_ℝ⟨ξ-η⟩^s/2f(ξ-η)⟨η⟩^s/2g(η) dηdξ
       ≲∫_ℝφ(η)ω(η/r) dηf_H^s/2g_H^s/2.

Hence A_r≲∫_ℝφ(η)ω(η/r) dη
and this last integral tends to zero by
the dominated convergence theorem as r→∞,
since ω is bounded above by a polynomial and
lim_η→ 0ω(η)→ 0.

Similarly, for f,g∈ L^2(ℝ) we have

    ⟨ B_rf, g⟩
    =
    1/2π|
    ∫_ℝ∫_ℝφ_r(η)f(ξ-η)(m^-1/2(ξ)-m^-1/2(ξ-η))g(ξ) dηdξ|
    
    
    =
    1/2π|
    ∫_ℝ∫_ℝφ_r(η)f(ξ-η)(m(ξ-η)-m(ξ)/m^1/2(ξ-η)m^1/2(ξ)(m^1/2(ξ-η)+m^1/2(ξ)))g(ξ) dηdξ|
    
    
    ≲∫_ℝφ_r(η)ω(η)∫_ℝ⟨ξ-η⟩^s/2/m^1/2(ξ-η)f(ξ-η)⟨η⟩^s/2/m^1/2(η)g(η) dηdξ
    
    ≲∫_ℝφ(η)ω(η/r) dηf_L^2g_L^2,



and we can conclude in the same way as before that
B_r→ 0 as r→∞.









§ NEAR MINIMIZERS





In this section we provide necessary estimates for
the infimum

    I_q = inf _u ∈ U_qℰ(u)

and for those u ∈ U_q that give values ℰ(u)
close to this infimum.




The regarded functional (<ref>)
is non-negative and so the same is true for the infimum.
However, we also need an upper bound for I_q
and this is addressed in the next result. 


There exist constants D, q_0>0 such that for q∈ (0,q_0)

    0 ⩽ I_q < m(0)q - Dq^1+β,

with β=s'/2s'-1.



It is immediate that 0 ⩽ I_q.
To establish the other inequality we consider φ∈ C^∞(ℝ), with supp(φ̂)⊆(-1,1), φ(x)≤ 0, x∈ℝ and 𝒬(φ)=1. We rescale and define φ_q,α(x)=√(q/α)φ(x/α), α>1, so that 𝒬(φ_q,α)=q.

We first note that

    ℒ(φ_q,α)≤ m(0)q+C_1qα^-s', C_1>0,

and using Proposition <ref>

    𝒩_r(φ_q,α)≤ C_2q^2, C_2>0.

In order to estimate 𝒩_c(ϕ_q,α) we begin by estimating

    0≤ m^1/2(ξ)-m^1/2(0)   ≤m(ξ)-m(0)/2√(m(0)),
    m^-1/2(ξ)-m^-1/2(0)   ≤m(ξ)-m(0)/2m(0)√(m(0)),

and then, using Lemma <ref>, we find that

    𝒩_2c(φ_q,α)   ≲ q^3/2α^-s'-1/2,
    𝒩_3c(φ_q,α)   ≲ q^3/2α^-s'.

Moreover, since φ(x)≤ 0, we have that

    1/2√(m(0))∫_ℝφ_q,α(x)^3 dx   =-2C_0q^3/2α^-1/2, C_0>0
    𝒩_1c(φ_q,α)   ≤ 0.

Hence, it follows from the above estimates that there exists α_0>1, such that for α≥α_0,

    𝒩_c(φ_q,α)≤ -C_0q^3/2α^-1/2,

and combining this with (<ref>), (<ref>), yields

    ℰ(φ_q,α)≤ m(0)q-(C_0q^3/2α^-1/2-C_1qα^-s')+C_2q^2,

and by choosing α^-s'=Bq^β, with 0<B≤α_0^-s'q^-β, so that α≥α_0, we get from (<ref>) that

    ℰ(φ_q,α)≤ m(0)q-(C_0B^1/(2s')-C_1B)_=:2Dq^1+β+C_2q^2,

By choosing B small enough we have that D>0, and if we in addition choose q_0 sufficiently small, we find that

    I_q≤ℰ(φ_q,α)< m(0)q-Dq^1+β.




Note that it is possible to weaken the restriction
on s' in the last proposition, namely
imposing s' > 1/2 instead.
It can be done by more accurate estimate of
the term 𝒩_r(φ_q,α).
In other words it is not important here that β < 1.
However, it will be important for the subadditivity of I_q
below.





I have added this unimportant remark
since we actually use β(s') < 1
in proposition <ref> only.








We now define a near minimizer to be an element u of U_q such that

    ℰ(u) < m(0)q - Dq^1+β.

By the previous proposition, there exist such elements u ∈ U_q. 

A near minimizer u ∈ U_q satisfies

    u_H^s/2^2 ≲ q.



Using propositions  <ref>, <ref> and <ref>, we find that 

    u_H^s/2(ℝ)^2   ≃ℒ(u)
       =ℰ(u)-𝒩(u)
       ≲ m(0)q-Dq^1+β+u_L^2(ℝ)^2u_H^s/2(ℝ)+u_L^2(ℝ)^4
       ≲ m(0)q-Dq^1+β+qu_H^s/2(ℝ)+q^2.

Hence, it follows that for q sufficiently small

    u_H^s/2(ℝ)^2≲ m(0)q-Dq^1+β≲ q.


We next show that I_q is strictly subadditive as a function of q.
This is essential when proving that dichotomy cannot occur. 

For any q_1,q_2∈(0,q_0) such that q_1+q_2∈(0,q_0),
holds

    0<I_q_1+q_2<I_q_1+I_q_2.



We show that I_q is strictly subhomogeneous, i.e

    I_aq<aI_q,  a>1, q<aq<q_0,

from which the strict subadditivity follows from a standard argument.
First we show that (<ref>) holds for a∈(1,2].
Let {u_n}_n=1^∞ be a minimizing sequence. From (<ref>) we have that

    ℒ(u_n)+𝒩_c(u_n)+𝒩_r(u_n)<m(0)q-Dq^1+β,

and since ℒ(u_n)≥ m(0)q, 𝒩_r(u)≥ 0, we get from (<ref>) that

    𝒩_c(u)<-Dq^1+β.

We also note that √(a)-1≥ (a-1)/(1+√(2)). With this in mind we see that

    I_aq   ≤ℰ(a^1/2u_n)
       =ℒ(a^1/2u_n)+𝒩(a^1/2u_n)
       =aℒ(u_n)+a^3/2𝒩_c(u_n)+a^2𝒩_r(u_n)
       =aℰ(u_n)-a(𝒩_c(u_n)+𝒩_r(u_n))+a^3/2𝒩_c(u_n)+a^2𝒩_r(u_n)
       =aℰ(u_n)+(a^3/2-a)𝒩_c(u_n)+(a^2-a)𝒩_r(u_n)
       ≤ aℰ(u_n)-(a^3/2-a)Dq^1+β+(a^2-a)C_3q^2
       ≤  aℰ(u_n)-(a^2-a)(Dq^1+β/1+√(2)-C_3q^2).

Hence, for q_0  sufficiently small

    I_aq+(a^2-a)Dq^1+β/2√(2)<aI_q,

which implies (<ref>) for a∈(1,2], but also that I_q>0, for q∈(0,q_0), proving the first inequality in (<ref>). For the general case when a>1, we choose l∈ℕ sufficiently big so that a∈(1,2^l]. Then a^1/l∈(1,2], and so

    I_aq=I_a^1/la^(l-1)/lq<a^1/lI_a^(l-1)/lq=a^1/lI_a^1/la^(l-2)/lq<a^2/lI_a^(l-2)/lq<…<aI_q.







§ EXISTENCE OF MINIMIZERS




In order to establish the existence of minimizers, we will apply the concentration-compactness principle (Theorem <ref>) to e_n=u_n^2, where {u_n}_n=1^∞ is a minimizing sequence. The idea is to show that the vanishing and dichotomy scenarios cannot occur and then prove the existence of a minimizer using concentration. We start by excluding the vanishing scenario.

Vanishing does not  occur. 





	Let {u_n}_n=1^∞⊆ U_q be a minimizing sequence of ℰ. We point out here that since L^-1/2u_n∈ H^s(ℝ) and s>1/2, we have that L^-1/2u_n∈ L^∞(ℝ).
	
	

	By Lemma <ref> we have
	
    | 𝒩_c(u) | ≲‖ u ‖_L^2 (ℝ)^2
    		‖ L^-1/2u ‖_L^∞ (ℝ)

	and so for a minimizing sequence
	
    q^β≲‖ L^-1/2u_n ‖_L^∞ (ℝ)
    		,

where we used (<ref>).
	Arguing as in the proof of <cit.>, we have for any x ∈ℝ that
	
	
    ‖ L^-1/2u_n ‖_L^∞ (x - 1, x + 1)≲‖ L^-1/2u_n ‖_L^2 (x - 1, x + 1)
    		^1 - 1/(2s)‖ L^-1/2u_n ‖_H^s (ℝ)
    		^1/(2s)
    ≲‖ L^-1/2u_n ‖_L^2 (x - 1, x + 1)
    		^1 - 1/(2s)‖ u_n ‖_H^s/2 (ℝ)
    		^1/(2s)≲
    		q^1/(4s)‖ L^-1/2u_n ‖_L^2 (x - 1, x + 1)
    		^1 - 1/(2s),

	
	and hence
	
    q^β - 1/(4s)≲sup_x ∈ℝ‖ L^-1/2u_n ‖_L^2 (x - 1, x + 1)
    		^1 - 1/(2s),

	
	which means that L^-1/2u_n cannot vanish.
	
	Now we  show that L^-1/2u_n is vanishing if
	one assumes that u_n is vanishing.
	In order to do this we start by decomposing

    (L^-1/2u_n)(x)   =(ℱ^-1(m^-1/2) * u_n)(x)
       =∫_ℝℱ^-1(m^-1/2)(y)u_n(x-y) dy
       =∫_y<ϵℱ^-1(m^-1/2)(y)u_n(x-y) dy_=:I_1+∫_ϵ≤y≤ Rℱ^-1(m^-1/2)(y)u_n(x-y) dy_=:I_2
         +∫_y≥ Rℱ^-1(m^-1/2)(y)u_n(x-y) dy_=:I_3,

and so

    L^-1/2u_n_L^2(x̃-1,x̃+1)≤I_1_L^2(x̃-1,x̃+1)+I_2_L^2(x̃-1,x̃+1)+I_3_L^2(x̃-1,x̃+1).

The goal is then to show that each of the above integrals can be made arbitrarily small.


By assumption there exists
p ∈ (1, 2) ∩ [2 / (s+1), 2)
such that (<ref>) holds, and so‖ℱ^-1(m^-1/2)
	‖ _L^p(- ε, ε)
	= o(1)
as ε→ 0.
On the other hand its dual number p' satisfies
condition
1/2 - 1/p' ⩽ s/2
resulting in the embedding
H^s/2(ℝ) ↪ L^p'(ℝ)
	.
Thus applying Hölder's inequality to I_1 yields

    I_1_L^2(x̃-1,x̃+1)^2
    	⩽∫ _x̃-1^x̃+1‖ℱ^-1(m^- 1/2)
    	‖ _L^p(- ε, ε)^2
    	‖ u_n ‖ _L^p'(ℝ)^2
    	 dx
    	= o(1)
    	ε→ 0.


For  I_3 we apply the Cauchy–Schwarz inequality
as follows


    I_3_L^2(x̃-1,x̃+1)^2
    	⩽∫ _x̃-1^x̃+1‖ℱ^-1(m^- 1/2)
    	‖ _L^2( ℝ∖ (- R, R) )^2
    	‖ u_n ‖ _L^2(ℝ)^2
    	 dx
    	= o(1)
    	 R →∞.


After choosing ε, R we turn our attention
to I_2


    I_2_L^2(x̃-1,x̃+1)^2
    	⩽∫ _x̃-1^x̃+1‖ℱ^-1(m^- 1/2)
    	‖ _L^2( (- R, R) ∖ (- ε, ε) )^2
    	‖ u_n(x - y) ‖ _L^2( ε < |y| < R )^2
    	 dx
    	
    ⩽
    	C(ε, R)
    	‖ u_n ‖ _L^2( -1 - |x̃| - R , 1 + |x̃| + R )^2
    	→ 0
    	 n →∞,


if one assumes vanishing of u_n.


We next turn our attention to the dichotomy scenario.

Dichotomy cannot occur.


Let χ :ℝ→ [0,1] be a smooth cutoff function with χ(x)=1, for x≤ 1 and χ(x)=0, for x≥ 2, and such that

    χ=χ_1^2,   1-χ=χ_2^2,

where χ_1,χ_2 are smooth. Next, let w_n(x)=u_n(x-x_n) and 

    w_n^(1)(x)=χ_1(x/M_n)_=:χ_1n(x)w_n(x),   w_n^(2)(x)=χ_2(x/M_n)_=:χ_2n(x)w_n(x),

Note that from the dichotomy assumption

    1/2∫_M_n≤x≤ 2M_nw_n^2 dx   ≤1/2∫_M_n≤x≤ N_nw_n^2 dx
       =1/2∫_-N_n^N_nw_n^2 dx-1/2∫_-M_n^M_nw_n^2 dx
       → q^*-q^*
       =0.

Since w_n^i(x)≤w_n(x), i=1,2, it follows directly that ∫_M_n≤x≤ 2M_n(w_n^(i))^2 dx→ 0, as n→∞.
From this we can then deduce

    1/2∫_ℝ(w_n^(1))^2 dx=1/2∫_-M_n^M_nw_n^2 dx-1/2∫_M_n≤x≤ 2M_n(w_n^(1))^2 dx→ q^*,

and similarly 

    1/2∫_ℝ(w_n^(2))^2 dx=1/2∫_ℝw_n^2 dx-1/2∫_-2M_n^2M_nw_n^2 dx+1/2∫_M_n≤x≤ 2M_n(w_n^(2))^2 dx→ q-q^*.

We next show that

    ℰ(w_n^(1))+ℰ(w_n^(2))-ℰ(w_n)→ 0, n→∞.

As a first step towards this, we show that

    ℒ(w_n^(1))+ℒ(w_n^(2))-ℒ(w_n)→ 0,  n→∞

Indeed, note that

    ℒ(w_n^(1))+ℒ(w_n^(2))-ℒ(w_n)=1/2∫_ℝw_n^(1)Lw_n^(1)+w_n^(2)Lw_n^(2)-(χ_1n^2+χ_2n^2)w_nLw_n dx,

and using Lemma <ref> we find that

    ∫_ℝw_n^(1)Lw_n^(1)-χ_1n^2w_nLw_n dx   =∫_ℝχ_1nw_n(L(χ_1nw_n)-χ_1nLw_n) dx
       =∫_ℝχ_1nw_n[L,χ_1n] dx
       → 0,  n→∞

In the same way we find that

    ∫_ℝw_n^(2)Lw_n^(2)-χ_2n^2w_nLw_n dx=∫_ℝχ_2nw_n[L,χ_2n-1]w_n dx→ 0,  n→∞,

hence, (<ref>) holds.
The next step is to show that

    𝒩(w_n^(1))+𝒩(w_n^(2))-𝒩(w_n)→ 0,  n→∞,

and for this we use the decomposition 𝒩=𝒩_c+𝒩_r, and show that

    𝒩_c(w_n^(1))+𝒩_c(w_n^(2))-𝒩_c(w_n)   → 0  n→∞,
    𝒩_r(w_n^(1))+𝒩_r(w_n^(2))-𝒩_r(w_n)   → 0  n→∞.

Starting with (<ref>), we note that

    𝒩_c(w_n^(1))+𝒩_c(w_n^(2))-𝒩_c(w_n)   =1/2∫_ℝ(L^1/2w_n^(1)(L^-1/2w_n^(1))^2+L^1/2w_n^(2)(L^-1/2w_n^(2))^2
           -(χ_1n^2+χ_2n^2)L^1/2w_n(L^-1/2w_n)^2) dx,

furthermore

    ∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1))^2-χ_1n^2L^1/2w_n(L^-1/2w_n)^2 dx
         =∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1))^2-χ_1n^2L^1/2w_n^(1)(L^-1/2w_n)^2 dx
           +∫_ℝχ_1n^2L^1/2w_n^(1)(L^-1/2w_n)^2-χ_1n^2L^1/2w_n(L^-1/2w_n)^2 dx,

and using Lemma <ref> we find that

    ∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1))^2-χ_1n^2L^1/2w_n^(1)(L^-1/2w_n)^2 dx
       =∫_ℝL^1/2w_n^(1)((L^-1/2w_n^(1))^2-χ_1n^2(L^-1/2w_n)^2) dx
       =∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1)+χ_1nL^-1/2w_n)(L^-1/2w_n^(1)-χ_1nL^-1/2w_n) dx
       =∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1)+χ_1nL^-1/2w_n)[L^-1/2,χ_1n]w_n dx
       → 0,  n→ 0,

and

    ∫_ℝχ_1n^2L^1/2w_n^(1)(L^-1/2w_n)^2-χ_1n^2L^1/2w_n(L^-1/2w_n)^2 dx
    
       =∫_ℝχ^2_1nL^1/2(w_n^(1)-w_n)(L^-1/2w_n)^2 dx
       =∫_ℝL^1/2(χ_1n(w_n^(1)-w_n))χ_1n(L^-1/2w_n)^2 dx
         -∫_ℝ[L^1/2,χ_1n](w_n^(1)-w_n)χ_1n(L^-1/2w_n)^2 dx,

where

    |
    ∫_ℝL^1/2(χ_1n(w_n^(1)-w_n))χ_1n(L^-1/2w_n)^2 dx
    |
    
       =
    |
    ∫_ℝχ_1n(χ_1n-1)w_nL^1/2(χ_1n(L^-1/2w_n)^2) dx
    |
    
       ≤χ_1n(χ_1n-1)w_n_L^2L^1/2(χ_1n(L^-1/2w_n)^2)_L^2
       ≲w_n_L^2([-2M_n,-M_n]∪[M_n,2M_n])w_n_H^s/2^2
       → 0,  n→∞,

and ∫_ℝ[L^1/2,χ_1n](w_n^(1)-w_n)χ_1n(L^-1/2w_n)^2 dx→ 0,  n→∞, according to Lemma <ref>.
Hence lim_ n→∞∫_ℝL^1/2w_n^(1)(L^-1/2w_n^(1))^2-χ_1n^2L^1/2w_n(L^-1/2w_n)^2 dx= 0, and in the same way we can show that lim_ n→∞∫_ℝL^1/2w_n^(2)(L^-1/2w_n^(2))^2-χ_2n^2L^1/2w_n(L^-1/2w_n)^2 dx= 0, which implies (<ref>). The limit (<ref>) can be shown using similar techniques as (<ref>) and we therefore omit the details.

We conclude that (<ref>) holds, which together with (<ref>) implies (<ref>). Since {w_n}_n=1^∞ is a minimizing sequence, we get that

    lim_n→∞ℰ(w_n^(1))+ℰ(w_n^(1))→ I_q.

However,

    lim_n→∞(
    	ℰ(v_n^(i))
    	-
    	ℰ(w_n^(i))
    )
    =0,  i=1,2,

where v_n^(1)=√(q^*/Q(w_n^(1)))w_n^(1),  v_n^(2)=√((q-q^*)/Q(w_n^(2)))w_n^(2). By construction v_n^(1)∈ U_q^*, v_n^(2)∈ U_q-q^*, and so using (<ref>), (<ref>), we find that 

    I_q= lim_n→∞ℰ(v_n^(1))+ℰ(v_n^(1))≥ I_q^*+I_q-q^*,

which contradicts Proposition <ref>.







	There exists u ∈ U_q solving minimization problem
	ℰ(u) = I_q.






By the concentration-compactness principle our
minimizing sequence
e_n = u_n^2 ∈ L^1 (ℝ)
	,
n ∈ℕ,
concentrates.
Moreover, due to the translation invariance
one can assume that it concentrates around zero,
and so

    ∫ _|x| > r u_n^2(x) dx →  0
    	
    	n ∈ℕ r →∞
    	.

In addition, {u_n}_n=1^∞ is a bounded sequence in H^s/2(ℝ) due to Proposition <ref>,
and so

    ‖ (u_n)_h - u_n ‖ _L^2^2
    	≲
    	q ‖ξ↦ | e^iξ h - 1 |
    		⟨ξ⟩ ^-s/2‖
    	_L^∞^2

that tends to zero uniformly with respect to
n ∈ℕ as h → 0.
Taking into account the boundedness of
{u_n}_n=1^∞ in L^2(ℝ) one deduces from
the Frechet–Kolmogorov theorem
that {u_n}_n=1^∞ is relatively compact in L^2(ℝ).
Thus we can assume that {u_n}_n=1^∞ converges to some u in L^2(ℝ).
Again using that {u_n}_n=1^∞ is bounded in H^s/2(ℝ) , we may in addition assume that u_n
converges weakly in H^s/2(ℝ) to u.
Hence u ∈ U_q and it is left to check
that it solves the minimization problem.


Firstly, applying the weak lower semi-continuity argument
we deduce

    ℒ(u) ⩽lim inf _n →∞ℒ(u_n).

Indeed, the square root of ℒ(u) defines a norm
in H^s/2(ℝ), equivalent to the standard Sobolev norm.
By the Mazur theorem a closed ball is weakly closed.
The latter property implies the weak lower semi-continuity
of the functional ℒ.


It is left to show that 𝒩(u_n)
tends to 𝒩(u) as n →∞.
The cubic part is estimated as

    | 𝒩_c(u) - 𝒩_c(u_n) |
    	⩽1/2|
    		∫_ℝ( L^-1/2 u )^2 L^1/2 (u- u_n)
    	 dx|
    	
    
    	+
    	1/2|
    		∫_ℝ(
    			( L^-1/2 u )^2
    			-
    			( L^-1/2 u_n )^2
    		)		
    		L^1/2 u_n
    	 dx|
    	=
    	1/2|
    		∫_ℝ (u- u_n) L^1/2( L^-1/2 u )^2
    	 dx|
    	
    
    	+
    	1/2|
    		∫_ℝ(
    			L^-1/2 (u - u_n)
    		)		
    		(
    			L^-1/2 (u + u_n)
    		)		
    		L^1/2 u_n
    	 dx|
    	≲‖ u - u_n ‖ _L^2‖
    		L^1/2( L^-1/2 u )^2
    	‖ _L^2
    
    	+
    	‖
    		L^-1/2 (u - u_n)
    	‖ _H^s/2‖
    		L^-1/2 (u + u_n)
    	‖ _H^s/2‖
    		L^1/2 u_n
    	‖ _L^2≲ q
    	‖ u - u_n ‖ _L^2


which tends to zero as n →∞.
For the remainder we have


    | 𝒩_r(u) - 𝒩_r(u_n) |
    	
    
    	=
    	1/8|
    		∫_ℝ(
    			L^-1/2 (u - u_n)
    		)		
    		(
    			L^-1/2 (u + u_n)
    		)		
    		(
    			( L^-1/2 u )^2
    			+
    			( L^-1/2 u_n )^2
    		)		
    	 dx|
    	
    ≲‖
    		L^-1/2 (u - u_n)
    	‖ _H^s/2‖
    		L^-1/2 (u + u_n)
    	‖ _H^s/2(
    		‖
    			L^-1/2 u
    		‖ _L^4^2
    		+
    		‖
    			L^-1/2 u_n
    		‖ _L^4^2
    	)
    	
    ≲ q ^3/2‖ u - u_n ‖ _L^2

that tends to zero as n →∞.
Summing up we obtain

    I_q ⩽ℰ(u)
    	⩽lim inf _n →∞ℰ(u_n) = I_q

which concludes the proof.




We finish the proof of Theorem <ref> by proving the estimate. Let u be a minimizer. We know that u satisfies the Euler–Lagrange equation

    λ u+dℰ(u)=0.

Taking the inner product in this equation with u yields

    -2λ q   =dℰ(u)(u)
       =2ℒ(u)+3𝒩_c(u)+4𝒩_r(u)
       =-ℒ(u)+3ℰ(u)+4𝒩_r(u)

Since ℒ(u)≥ m(0) q and 𝒩_c(u)=𝒪(q^3/2), 𝒩_r(u)=𝒪(q^2) by Proposition <ref>, it is easy to see from the second inequality in (<ref>) that for q sufficiently small 

    -λ>m(0)/2.

For the upper bound we use (<ref>) together with propositions <ref>, <ref>, <ref> to deduce that

    -2λ q   =-ℒ(u)+3ℰ(u)+4𝒩_r(u)
       =-ℒ(u)+3I_q+4𝒩_r(u)
       ≤ -m(0)q+3(m(0)q-Dq^1+β)+𝒪(q^2)
       =2m(0)q-3Dq^1+β+𝒪(q^2),

hence, for q sufficiently small

    -λ<m(0)-Dq^β.





§ LONG WAVE APPROXIMATION




In this section we return to the initial variational
problem for the Whitham–Boussinesq system.
So from now on L = K defined by (<ref>).
We point out that all calculations below are also
valid for L = 1 - ∂_x^2 / 3 as well.
We will show that all minimizers are infinitely smooth
and refine existing estimates for them.






	There exists q_0 > 0 such that for each r ⩾ 0
	holds
	‖ u ‖ _H^r^2 ≲ q
	uniformly for q ∈ (0, q_0) and u ∈ D_q.





Firstly, one can notice that the statement holds for r ∈ [0, 1/2], due to Proposition <ref>.
We will extend the result by induction to bigger values of r
applying Formula (<ref>).


Let r ⩾ 1/2, then from the equivalence
of operators K, J and product estimates in Sobolev spaces
we deduce

    ‖
    		K^-1/2( K^-1/2 v ) ^3
    	‖  _H^r≲‖ v ‖  _H^r^3
    	,



    ‖
    		K^-1/2( (K^1/2 v)( K^-1/2 v) )
    	‖  _H^r≲‖ v ‖  _H^r^2
    	,



    ‖
    		K^1/2( K^-1/2 v ) ^2
    	‖  _H^r≲‖ v ‖  _H^r^2

for any v ∈ H^r(ℝ).
All three constants here depend only on r.


Now for any minimizer u ∈ D_q
calculate Ku by Formula (<ref>) and
obtain

    ‖ u ‖  _H^r + 1≲‖ Ku ‖  _H^r⩽
    	| λ | ‖ u ‖  _H^r
    	+ 1/2‖
    		K^-1/2( K^-1/2 u ) ^3
    	‖  _H^r
    
    	+
    	‖
    		K^-1/2(( K^1/2 u)( K^-1/2 u) )
    	‖  _H^r
    	+ 1/2‖
    		K^1/2( K^-1/2 u ) ^2
    	‖  _H^r≲√(q)


for any r ⩾ 1/2.
We have used |λ| ⩽ 1
according to Theorem <ref>.
This concludes the proof by induction.






	There exist q_0 > 0 and C > 0 such that
	the following estimates hold
	
	
    ‖ u ‖ _L^∞⩽ Cq^2/3
    		,

	
	
    ‖∂_x u ‖ _L^2^2 ⩽ Cq^5/3
    		,

	
	
    ‖∂_x^2 u ‖ _L^2^2 ⩽ Cq^7/3

	uniformly for q ∈ (0, q_0) and u ∈ D_q.



Introducing the notation

    M(u) = 
    	1/2 K^-1/2( K^-1/2 u ) ^3
    	+ K^-1/2(( K^1/2 u)( K^-1/2 u) )
    	+ 1/2 K^1/2( K^-1/2 u ) ^2

one can rewrite Equation (<ref>) in the form

    ( λ + K ) u = - M(u)
    	.

Note that
- λ∈( 0, 1 - Dq^2/3)
according to Theorem <ref>
and so λ + 1 > Dq^2/3.
The Fourier transform of minimizer u can be estimated as

    |
    		u(ξ)
    	|
    	=
    	|
    		ℱ( M(u) ) /λ + m(ξ) |
    	⩽ | ℱ( M(u) ) | / Dq^2/3 + m(ξ) - 1 ≲
    	| ℱ( M(u) )(ξ) |
    	(
    		χ _ |ξ| ⩽ 1 (ξ) / q^2/3 + ξ^2 
    		+	
    		χ _ |ξ| > 1 (ξ) / q^2/3 + |ξ| )

where χ_A(ξ)
stands for the characteristic function of a set A.
As was shown in the proof of Lemma <ref>
M(u), is smooth and its H^s-norm is bounded by q
for any non-negative s.
Hence ℱ( M(u) ) multiplied by any
power of ξ is bounded by q
with respect to L^2-norm.


Let us show that the L^∞-norm of
ℱ( M(u) ) is bounded by q.
Indeed, we have

    |
    		ℱ(
    			K^1/2( K^-1/2 u ) ^2
    		)
    		(ξ)
    	|
    	≲∫_ℝ√( m(ξ) ) | u(ξ - ζ) u(ζ) | /√( m(ξ - ζ) m(ζ) ) dζ≲‖ u ‖ _L^2^2
    	≲
    	q
    	,



    |
    		ℱ(
    			K^-1/2(( K^1/2 u)( K^-1/2 u) )
    		)
    		(ξ)
    	|
    	≲∫_ℝ√( m(ξ - ζ) ) | u(ξ - ζ) u(ζ) | /√( m(ξ) m(ζ) ) dζ≲‖ u ‖ _L^2^2
    	≲
    	q


and similarly

    |
    		ℱ(
    			K^-1/2( K^-1/2 u ) ^3
    		)
    		(ξ)
    	|
    	≲‖ u ‖ _L^2‖( K^-1/2 u ) ^2 ‖ _L^2≲‖ u ‖ _L^2^3
    	≲
    	q^3/2
    	.


Thus
‖ℱ( M(u) ) ‖ _L^∞≲ q
	.
So we are in a position to prove (<ref>),
indeed,


    ‖ u ‖ _L^∞≲‖u‖ _L^1≲∫ _ |ξ| ⩽ 1  | ℱ( M(u) )(ξ) | / q^2/3 + ξ^2  dξ
    	+	
    	∫ _ξ>1  | ℱ( M(u) )(ξ) |/q^2/3+ |ξ|  dξ
    ≲
    	q^-1/3‖ℱ( M(u) ) ‖ _L^∞
    	+
    	‖ℱ( M(u) ) ‖ _L^2≲
    	q^2/3
    	.


Estimate (<ref>) is proved as follows


    ‖∂_x u ‖ _L^2^2
    	=
    	‖ξ↦ξu(ξ) ‖ _L^2^2
    	≲∫ _ |ξ| ⩽ 1 ξ^2 | ℱ( M(u) )(ξ) |^2 / ( q^2/3 + ξ^2 )^2  dξ
    	+	
    	∫ _ξ> 1 ξ^2| ℱ( M(u) )(ξ) |^2/ (q^2/3+|ξ|)^2 dξ
    ≲
    	q^-1/3‖ℱ( M(u) ) ‖ _L^∞^2
    	+
    	‖ℱ( M(u) ) ‖ _L^2^2
    	≲
    	q^5/3
    	.


A straightforward repetition of the last argument
for the second derivative of the minimizer
gives


    ‖∂_x^2 u ‖ _L^2^2
    	=
    	‖ξ↦ξ^2 u(ξ) ‖ _L^2^2
    	≲∫ _ |ξ| ⩽ 1 ξ^4 | ℱ( M(u) )(ξ) |^2 / ( q^2/3 + ξ^2 )^2  dξ
    	+	
    	∫ _ξ>1 ξ^4| ℱ( M(u) )(ξ) |^2 /(q^2/3+ |ξ|)^2  dξ
    ≲
    	q^1/3‖ℱ( M(u) ) ‖ _L^∞^2
    	+
    	‖ℱ( ∂_x M(u) ) ‖ _L^2^2
    	≲
    	q^7/3
    	+
    	‖∂_x M(u) ‖ _L^2^2


that is only 𝒪(q^2) and so weaker than (<ref>). 
However, Estimate (<ref>)
is a refinement compared with Lemma <ref>, so it can be used for more delicate estimate
of the square norm
‖∂_x M(u) ‖ _L^2^2
as follows


    ‖∂_x K^1/2( K^-1/2 u ) ^2
    	‖ _L^2≲
    		K^-1/2 u  K^-1/2∂_x u
    	 _H^1/2
    ≲
    		K^-1/2 u
    	 _H^1
    		K^-1/2∂_x u
    	 _H^1/2≲
    	q^4/3
    	,



where product estimates were used.
To continue, first note that the estimate of the
derivative (<ref>), will not be spoiled if one
changes L^2-norm to H^s-norm with any s ⩾ 0.
In other words,
‖∂_x u ‖ _H^s≲ q^5/6
	,
and so


    ‖∂_x K^-1/2( K^1/2 u K^-1/2 u )
    	‖ _L^2≲
    		K^1/2∂_x u  K^-1/2 u
    	 _L^2
    	+
    	
    		K^1/2 u  K^-1/2∂_x u
    	 _L^2
    ≲‖∂_x u
    	‖ _H^3/2‖
    		u
    	‖ _L^2
    	+
    	‖
    		u
    	‖ _H^3/2‖∂_x u
    	‖ _L^2≲
    	q^4/3
    	.


The last remaining term is estimated similarly 


    ‖∂_x K^-1/2( K^-1/2 u ) ^3
    	‖ _L^2≲‖
    			( K^-1/2 u )^2 K^-1/2∂_x u
    	‖ _L^2≲‖
    		u
    	‖ _H^1/2 ^2
    	‖∂_x u
    	‖ _L^2≲
    	q^11/6
    	.


Thus

    ‖∂_x M(u) ‖ _L^2≲ q^4/3

that together with (<ref>)
conclude the proof of Estimate (<ref>).







	Lemmas <ref>,
	<ref>
	remain valid with the surface elevation
	η_u and velocity v_u defined by (<ref>), (<ref>)





	substituted instead of the minimizer u ∈ D_q.




We now turn to the task of approximating the solutions found in Theorem <ref> with solutions of the KdV-equation. For this part we follow <cit.> closely.


We introduce the long-wave scaling S_KdV(f)(x)=q^2/3f(q^1/3x) and note that when making the ansatz u=S_KdV(ψ) in (<ref>), the leading order part of the equation as q→ 0 is, with λ=-1+λ_0q^2/3,

    λ_0ψ+3/2ψ^2-ψ_xx/3=0.

Equation (<ref>) is the travelling wave version of the  KdV-equation, which has the up to translation the following unique solution

    ψ_KdV(x)
    = -λ_0^2(1/2√(3λ_0)x).

We note that (<ref>) is the Euler-Lagrange equation of the minimization problem 

    I_KdV:=min_ψ∈ V_1ℰ_KdV(ψ),

where 

    ℰ_KdV(ψ):=1/2∫_ℝψ_x^2/3+ψ^3 dx,

and V_1:={ψ∈ H^1(ℝ) : 𝒬(ψ)=1}. The constraint 𝒬(ψ_KdV)=1 requires that λ_0=3/16^1/3.
The relation between ℰ and ℰ_KdV is now established.

For u∈ H^2(ℝ) hold 

    ℰ(u)=𝒬(u)+ℰ_KdV(u)+ℰ_rem(u),

with

    ℰ_rem(u)   ≲∂_x^2u_L^2^2+u_L^∞∂_xu_L^2^2+u_L^2^2∂_xu_L^2+u_L^4^2∂_x^2u_L^2+u_L^2^4,
    ⟨dℰ_rem(u),u⟩   ≲∂_x^2u_L^2^2+u_L^∞∂_xu_L^2^2+u_L^2^2∂_xu_L^2+u_L^4^2∂_xu_L^2+u_L^2^4



We note that

    ℰ(u)   =1/2∫_ℝuKu dx+𝒩_c(u)+𝒩_r(u)
          =Q(u)+1/2∫_ℝu(K-1)u dx+1/2∫_ℝ u^3 dx+𝒩_1c(u)+𝒩_2cu+𝒩_3c(u)+𝒩_r(u)
    
         =Q(u)+ℰ_KdV(u)
    
           +1/2∫_ℝ(
    	m(ξ)-1-ξ^2/3) û^2 dξ+𝒩_1c(u)+𝒩_2c(u)+𝒩_3c(u)+𝒩_r(u)_=:ℰ_rem(u)

Since m(ξ)=ξ/tanh(ξ), we have that m(ξ)-1 - ξ^2/3≲ξ^4, so that

    ∫_ℝ|
    	m(ξ)- 1 - ξ^2/3|
    û^2 dξ≲∂_x^2u_L^2^2.

From Lemma <ref> we have

    𝒩_1c(u)≲∫_ℝu((K^-1/2-1)u)^2  dx≲u_L^∞∂_xu_L^2^2.

Similarly we find that

    𝒩_2c(u)   ≲u_L^4^2∂_xu_L^2,
    𝒩_3c(u)   ≲u_L^2^2∂_xu_L^2.

The term 𝒩_r(u) is estimated in Proposition <ref>, hence (<ref>) is established. The estimate (<ref>) is proved in a similar way and we therefore omit the details.


There exists q_0>0 such that 

    I_q   =q+ℰ_KdV(u)+𝒪(q^2), uniformly over u∈ D_q,
    
    I_q   =q+q^5/3I_KdV+𝒪(q^2).



Let u∈ D_q. From Lemma <ref> we know that u∈ H^r(ℝ) for any r≥ 0. In particular u∈ H^2(ℝ), hence by Lemma  <ref> 

    ℰ(u)=q+ℰ_KdV(u)+ℰ_rem(u).

Using (<ref>) together with Lemma <ref>, we get ℰ_rem(u)≲ q^2. Hence, (<ref>) follows.

Turning now to (<ref>) we let ψ=S_KdV^-1(u) and note that ψ∈ V_1 and

    ℰ_KdV(u)=q^5/3ℰ_KdV(ψ)≥ q^5/3I_KdV,

so this together with (<ref>) implies 

    I_q≥ q+q^5/3I_KdV+𝒪(q^2).

On the other hand, ũ:=S_KdV(ψ_KdV)∈ U_q,
so again using (<ref>) obtain

    I_q   ≤ℰ(ũ)
       =q+ℰ_KdV(ũ)+𝒪(q^2)
       =q+q^5/3ℰ_KdV(ψ_KdV)+𝒪(q^2)
       =q+q^5/3I_KdV+𝒪(q^2),

which concludes the proof of (<ref>).



The statement of Theorem <ref> is a summary of the following lemmas.


There exists q_0>0 such that for any
q∈ (0,q_0) and u∈ D_q there exists x_u∈ℝ such that 

    S^-1_KdV(u)-ψ_KdV(· -x_u)_H^1≲ q^1/6,

uniformly with respect to q∈(0,q_0) and u∈ D_q.

The proof of Lemma <ref>
is identical to the proof of <cit.>
and is therefore omitted.

We next relate the two Lagrange multipliers λ and λ_0.


The Lagrange multipliers related to the minimization problem (<ref>), satisfy 

    λ=-1+λ_0q^2/3+𝒪(q^5/6).




Let u∈ D_q.
From Lemma <ref> we have

    ⟨dℰ(u),u⟩ =2q+⟨dℰ_KdV(u),u⟩ +𝒪(q^2).

Moreover, ⟨dℰ_KdV(u),u⟩=q^5/3⟨dℰ_KdV(S_KdV^-1(u)),S_KdV^-1(u)⟩,
and by Lemmas <ref>, <ref>

    ⟨dℰ_KdV(S_KdV^-1(u)),S_KdV^-1(u)⟩-⟨dℰ_KdV(ψ_KdV),ψ_KdV⟩ =𝒪(q^1/6).

Combining this with (<ref>), we obtain

    ⟨dℰ(u),u⟩=2q+q^5/3⟨dℰ_KdV(ψ_KdV),ψ_KdV⟩+𝒪(q^11/6).

On the other hand,
from the Euler-Lagrange equations we have

    2λ q+⟨dℰ(u),u⟩   =0,
    
    2λ_0+⟨dℰ_KdV(ψ_KdV),ψ_KdV⟩   =0,

and when we combine this with (<ref>), we get

    -2λ q=2q-2λ_0q^5/3+𝒪(q^11/6),

and dividing with -2q yields

    λ=-1+λ_0q^2/3+𝒪(q^5/6).




For each solution u of (<ref>),
we have the corresponding physical parameters η_u, v_u
defined by (<ref>), (<ref>)




where -1/c^2=λ=-1+λ_0q^2/3+𝒪(q^5/6)
by Lemma <ref>.
We have the following estimates for η_u, v_u
that are similar to the one given in Lemma <ref>.


There exists q_0>0 such that for q∈ (0,q_0) and u∈ D_q there exists x_u∈ℝ such that 

    S^-1_KdV(η_u)+ψ_KdV(· -x_u)
    	 _H^1/2≲ q^1/6
    	,


    S^-1_KdV(v_u)+ψ_KdV(· -x_u)
    	 _H^3/2≲ q^1/6

uniformly with respect to q∈(0,q_0) and u∈ D_q.



We will prove the first inequality.
The second one can be proved analogously.
Firstly, one can notice that due to
1/2 < -λ < 1
in accordance with to Estimate (<ref>),
it is enough to prove

    λ S^-1_KdV(η_u)
    		+ λψ_KdV(· -x_u)
    	 _H^1/2≲ q^1/6,


where x_u is taken as in Lemma <ref>.
The first term under the norm in (<ref>)
has the form


    λ S_KdV^-1(η_u)
    	=
    	q^-2/3/2(K^-1/2u)^2(q^-1/3·)
    	+
    	q^-2/3(K^1/2u)(q^-1/3·)

where the first element of the sum is negligible
in view of the straightforward estimate

    (K^-1/2u)^2(q^-1/3·)_H^1/2≲ q.


The second element of the sum can be rewritten as follows.
We note that

    (K^1/2u)(q^-1/3x)=(K_q^1/2u(q^-1/3·))(x),

where we used K_q to denote the Fourier multiplier operator with symbol m(q^1/3·).
We then get that q^-2/3(K^1/2u)(q^-1/3·)=K_q^1/2S_KdV^-1(u). Using this, we find that

    q^-2/3(K^1/2u)(q^-1/3·)-ψ_KdV(·-x_u)
    =
    K_q^1/2S_KdV^-1(u)-ψ_KdV(·-x_u)
    
    
    =
    K_q^1/2(S_KdV^-1(u)-ψ_KdV(·-x_u))+(K_q^1/2-1)ψ_KdV(·-x_u)
    .


Here the last term is estimated as

    (K_q^1/2-1)ψ_KdV(·-x_u)_H^1/2≲ q^1/3ψ_KdV_H^3/2.

Finally, we have

    λ S^-1_KdV(η_u)
    		+ λψ_KdV(· -x_u)
    	 _H^1/2≲
    		K_q^1/2(S_KdV^-1
    		- ψ_KdV(·-x_u))
    	 _H^1/2
    
    	+
    	
    		(K_q^1/2-1)ψ_KdV(·-x_u)
    	 _H^1/2
    	+
    	
    		( 1 + λ ) ψ_KdV(·-x_u)
    	 _H^1/2
    	+ q^1/3

that gives (<ref>) by
Lemma <ref> and <ref>.









0.05in
Acknowledgments.

E.D. was supported by the
Norwegian Research Council.
D.N. was supported by an ERCIM `Alain Bensoussan' Fellowship
and by grant no. 250070 from the Research Council of Norway.








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