On the asymptotics of real solutions for the Painlevé I equation
Wen-Gao Long Jun Xia






On the asymptotics of real solutions for the Painlevé I equation
    Wen-Gao Long^1,      Jun Xia^2,[Corresponding author]
    
================================================================



^1School of Mathematics and Computational Science, Hunan University of Science and Technology,

Xiangtan, 411201, China,






^2School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, 

Guangzhou 510665, China,













§ ABSTRACT 
In this paper, we revisit the asymptotic formulas of real Painlevé I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert method, we improve the error estimates of the oscillatory type asymptotics and provide precise error estimates of the singular type asymptotics. We also establish the corresponding asymptotics for the associated Hamiltonians of real Painlevé I transcendents. In addition, two typos in the mentioned asymptotic behaviors in literature are corrected. 

2010 mathematics subject classification: 33E17; 34A30; 34E05; 34M55; 41A60
Keywords and phrases: The Painlevé I equation, asymptotic expansions, Riemann-Hilbert approach.



On the asymptotics of real solutions for the Painlevé I equation
    Wen-Gao Long^1,      Jun Xia^2,[Corresponding author]
    
================================================================

equationsection






§ INTRODUCTION

Painlevé equations are a group of six nonlinear second-order ordinary differential equations
that possess the Painlevé property. This property requires that the movable singularities of the solutions must be poles, not branch points or essential singularities.
In general, the solutions of these equations do not have explicit expressions. Consequently,
asymptotic analysis is a common tool to study the behavior of solutions for Painlevé equations.

In this paper, we are concerned with the Painlevé I equation

    y”(x)=6y^2(x)+x,

and we concentrate on the asymptotic behaviors as  x→-∞ of real solutions to (<ref>).
Using the method of dominant balance (cf. <cit.>), it is easy to see that there are two classes of behaviors

    y(x)∼(-x/6)^1/2,      or     
    y(x)∼ -(-x/6)^1/2,

as x→-∞. For the second case, one can further show that as x→-∞,

    y(x)=-(-x/6)^1/2+d/(-x)^1/8cos(4·24^1/4/5(-x)^5/4-5d^2/8ln(-x)+χ)
    +O(x^-5/8),

where d and χ are constants; see <cit.>. It should be noted that the asymptotic formula (<ref>) also appears in the NIST handbook of mathematical functions (see <cit.>).

Indeed, Holmes and Spence <cit.> proved that there exist exactly three types
of real solutions by studying
a boundary value problem for (<ref>). Subsequently, with the help of the isomonodromy approach <cit.>, Kapaev <cit.> obtained the following classification of asymptotic behaviors as x→-∞ of real Painlevé I transcendents, in terms of the Stokes multiplier s_2 (see Section <ref> for definition).

  
  * When |s_2|<1, we have
  
    y(x)∼-(-x/6)^1/2+d/(-24x)^1/8cos(4·24^1/4/5(-x)^5/4-5d^2/8ln(-x)+χ),

where

    {
    d^2   =-1/πln(1-|s_2|^2),
    χ   = s_2-(19/8ln2+5/8ln3)d^2-Γ(-id^2/2)
    +3π/4.
    .

  
  * When |s_2|=1, we have
  
    y(x)=y(x)+s_1-s_-1/4·24^1/4√(π)(-x)^-1/8
    e^-4·24^1/4/5(-x)^5/4[1+O(x^-5/4)],

where y(x)=(-x/6)^1/2[1+O(x^-5/2)] as x→-∞ is the solution to (<ref>) with the Stokes multipliers s_2=s_-2=i, s_-1=s_1=i/2.
  
  * When |s_2|>1, we have

    y(x)∼-(-x/6)^1/2+(-x)^1/2/√(6)/3sin^2(2·24^1/4/5(-x)^5/4
    +5q/8ln(-x)+ρ),

where

    {ρ   =1/2πln(|s_2|^2-1),
    σ   =1/2 s_2+(19/8ln2+5/8ln3)ρ
    +1/2Γ(1/2-iρ)-π/4.
    .



The type (A), (B) and (C) solutions are called oscillatory, separatrix and singular solutions, respectively. In <cit.>, with the Riemann-Hilbert approach, Kapaev found all separatrix (also known as tronquée) solutions of (<ref>). Recently, Deaño <cit.> extends Kapaev's results by calculating explicitly exponentially small corrections with respect to the tronquée solutions. However, to the best of our knowledge, the error estimate in asymptotic formula (<ref>) and the asymptotics for the Hamiltonians in case (A) and (C) have not been provided in the existing literature.

In this paper, by performing a Deift-Zhou steepest descent analysis <cit.> to the Riemann-Hilbert problem of (<ref>), we obtain the following asymptotics for the Painlevé I transcendent y(x) and its Hamiltonian ℋ(x) (see (<ref>) for definition).




If |s_2|<1, we have, as x→-∞,

    y(x)   =-(-x/6)^1/2+√(a)/(-24x)^1/8cos(4·24^1/4/5(-x)^5/4-5a/8ln(-x)+ϕ)
    +O(x^-3/4),
    ℋ(x)   =-4(-x/6)^3/2
    +a(-3x/2)^1/4+√(a)/(-24x)^3/8sin(4·24^1/4/5(-x)^5/4-5a/8ln(-x)+ϕ)
           +O(x^-1),

where

    {
    a   =-1/πln(1-|s_2|^2),
    ϕ   = s_2-(19/8ln2+5/8ln3)a-Γ(-ia/2)
    -π/4.
    .

If |s_2|>1, we have, as x→-∞,

    y(x)   =-(-x/6)^1/2+(-x)^1/2/√(6)/3sin^2(2·24^1/4/5(-x)^5/4
    +5b/8ln(-x)+ψ)+O(x^-3/4),
    ℋ(x)   =-4(-x/6)^3/2
    -b(-24x)^1/4-(-3x/2)^1/4(2·24^1/4/5(-x)^5/4+5b/8ln(-x)+ψ)
           +O(x^-1),

where the error terms hold uniformly for x bounded away from the singularities appearing in the denominator and

    {
    b   =1/2πln(|s_2|^2-1),
    ψ   =1/2 s_2+(19/8ln2+5/8ln3)b
    +1/2Γ(1/2-ib)+π/4.
    .







In the asymptotic formula (<ref>), we improve the error estimate O(x^-5/8) in (<ref>) to O(x^-3/4). In the asymptotic formula (<ref>), we provide a rigorous error estimate O(x^-3/4). Moreover, we find that the angle 3π/4 in (<ref>) and  -π/4 in (<ref>) should be corrected by -π/4 and π/4, respectively.
The first author (W.-G. Long) of this paper apologies for committing the same typos in the Introduction of his other papers <cit.>, even though these typos do not affect the correctness of the main results of these papers. In Figures <ref> and <ref>, we have verified (<ref>) and (<ref>) numerically in special cases.



One should note from <cit.> that the Stokes multipliers of the special Painlevé I transcendent with (y(0),y'(0))=(0,0) are s_k=2icos(2π/5), k=0, ± 1,± 2.
In Appendix <ref>, we calculate the Stokes multipliers of the special solution with (p,H)=(0,0) by using the complex WKB method and find that s_k=-2icos(π/5), k=0, ± 1,± 2, where p is the pole location of the solution and H is a free parameter in the Laurent series

    y(x)=1/(x-p)^2-p/10(x-p)^2-1/6(x-p)^3+H(x-p)^4+⋯.

With this result, we successfully verify our asymptotic formula (<ref>) numerically for this special solution.









In Figure <ref> and Figure <ref>, we numerically verify the asymptotic behaviors of the Hamiltonian ℋ(x) in (<ref>) and (<ref>), respectively.












The rest of the paper is organized as follows. In Section <ref>, we formulate the Riemann-Hilbert problem of the Painlevé I equation (<ref>). Due to the different case: (1) |s_2|<1, (2) |s_2|>1, we apply the Deift-Zhou approach to analyze the Riemann-Hilbert problem for Ψ(z,x) as x→-∞ in Section <ref> and <ref>, respectively. In the final section <ref>, we prove Theorem <ref>. For the convenience of the reader, we put the parabolic cylinder and Airy parametrix model used in the asymptotic analysis, and the computation of Stokes multipliers of a special solution to Painlevé I in the Appendix.








§ RIEMANN-HILBERT PROBLEM

In this section, we describe the Riemann-Hilbert (RH) problem of the Painlevé I equation (<ref>), which was first introduced in <cit.>.
We seek a 2×2 matrix-valued function Ψ(z,x) satisfying the following properties.


  * Ψ(z,x) is analytic for z∈ℂ∖Γ, where Γ=∪_k=-2^2Γ_k with

    Γ_0=ℝ_+,  Γ_3=e^π iℝ_+,  Γ_±1=e^±2π i/5ℝ_+,  Γ_±2=e^±4π i/5ℝ_+,

see Figure <ref> for an illustration.

  * Ψ(z,x) satisfies the jump relation

    Ψ_+(z,x)=Ψ_-(z,x)S_k,     z∈Γ_k,

   where the Stokes matrices {S_k} take the forms
   
    S_0=[   1   0; s_0   1 ],       S_1= [   1 s_1;   0   1 ],         S_2=[   1   0; s_2   1 ],
       S_3=[  0 -i; -i  0 ],     S_-1= [    1 s_-1;    0    1 ],     S_-2=[    1    0; s_-2    1 ].

  Here, the Stokes multipliers {s_k} satisfy the restricted conditions

    1+s_ks_k+1=-is_k+3,     s_k+5=s_k,     k∈ℤ.


  * Ψ(z;x) satisfies the following asymptotic behavior as z→∞
  
    Ψ(z,x)=z^σ_3/41/√(2)[  1  1;  1 -1 ][I+∑_k=1^∞Ψ_k(x)z^-k/2]
      e^(4/5z^5/2+xz^1/2)σ_3,

  where the branches of powers of z are chosen such that z∈(-π,π).






It should be mentioned that in the RH analysis throughout, we will use frequently the matrix notations

    σ_1=[ 0 1; 1 0 ],    σ_2=[  0 -i;  i  0 ],    σ_3=[  1  0;  0 -1 ].



From <cit.>, we obtain the following expressions for the Painlevé I transcendent y(x) and the associated Hamiltonian

    ℋ(x)=1/2[y'(x)]^2-2y(x)^3-xy(x).


Let Ψ_k:=Ψ_k(x), k=1,2 be the coefficient in the expansion (<ref>). We have
  
    Ψ_1=[ -ℋ(x)     0;     0  ℋ(x) ],    Ψ_2=1/2[ ℋ(x)^2   y(x);   y(x) ℋ(x)^2 ].

Hence, we have

    y(x)   =2Ψ_2,12=2Ψ_2,21,
    ℋ(x)   =-Ψ_1,11=Ψ_1,22,

where Ψ_k,ij denotes the (i,j) entry of Ψ_k.







By (<ref>), it follows that

    1+s_2s_-2=-is_0,     s_1=i-s_-2/1+s_2s_-2,     s_-1=i-s_2/1+s_2s_-2,

if 1+s_2s_-2≠0. For the real solutions of the Painlevé I equation, we further have (see <cit.>)

    s̅_2=-s_-2,

which implies from (<ref>) that

    s̅_0=-s_0,    s̅_1=-s_-1,

and

    -is_0=1+s_2s_-2=1-|s_2|^2=1-|s_-2|^2∈ℝ.










§ ASYMPTOTICS OF Ψ FOR |S_2|<1

In this section, we first perform the Deift-Zhou nonlinear steepest descent analysis to the RH problem for Ψ(z,x) with 0<|s_2|<1. Then, the reduced case s_2=0 will be considered in the end of this section.



 §.§ Scaling transformation

Assume in what follows that x<0. We define

    A(z)=|x|^-σ_3/8Ψ(|x|^1/2z,x).

Then, A(z) satisfies the following RH problem.


  * A(z) is analytic for z∈ℂ∖Γ.


  * A(z) satisfies the same jump relation as Ψ(z,x); see (<ref>).


  * As z→∞, we have
  
    A(z)=z^σ_3/41/√(2)[  1  1;  1 -1 ][I+∑_k=1^∞Ψ_k(x)(|x|^1/2z)^-k/2]
      e^|x|^5/4θ(z)σ_3,

  where
  
    θ(z)=4/5z^5/2-z^1/2.













 §.§ The g-function transformation

To normalize the behavior of A(z) at infinity, we introduce the g-function

    g(z)=4/5(z-2α)^3/2(z+3α),    α:=√(6)/6.

where the branch of (z-2α)^3/2 is taken such that (z-2α)∈(-π,π). It is clear that

    g(z)=θ(z)+2α/3z^-1/2+O(z^-3/2),     z→∞,

and g(z) has saddle point at z=-α. The sign of g(z) in the complex plane is depicted in Figure <ref>.






The next transformation is defined by

    B(z)=[                          1 2α/3|x|^5/4-Ψ_1,11|x|^-1/4;                          0                          1 ]
    A(z)e^-|x|^5/4g(z)σ_3.

We come to the following RH problem for B(z).


  * B(z) is analytic for z∈ℂ∖Γ.


  * B(z) satisfies the jump relation

    B_+(z)=B_-(z){   e^|x|^5/4g(z)σ_3S_ke^-|x|^5/4g(z)σ_3,       z∈Γ_k, k=±1,±2,
       -iσ_1,       z∈Γ_3,
       e^|x|^5/4g_-(z)σ_3S_0e^-|x|^5/4g_+(z)σ_3,       z∈(0,2α),
       e^|x|^5/4g(z)σ_3S_0e^-|x|^5/4g(z)σ_3,       z∈(2α,+∞).
    .



  * As z→∞, we have
  
    B(z)=[I+B_1/z+O(z^-2)]z^σ_3/41/√(2)[  1  1;  1 -1 ],

 where
 
    B_1,21=Ψ_1,11|x|^-1/4-2α/3|x|^5/4,
       B_1,22=(Ψ_2,11-Ψ_2,12)|x|^-1/2
    -2α/3Ψ_1,11|x|+2α^2/9|x|^5/2.









 §.§ Contour deformation

In view of Figure <ref>, we see from (<ref>) that the jumps on Γ_±2 do not tend to the identity matrix as x→-∞. To overcome this, we define the transformation

    C(z)=B(z){   e^|x|^5/4g(z)σ_3S_2^-1e^-|x|^5/4g(z)σ_3,        z∈Ω_2,
       e^|x|^5/4g(z)σ_3S_-2e^-|x|^5/4g(z)σ_3,        z∈Ω_-2,
       I,       elsewhere.
    .

Thus, we have the following RH problem for C(z).


  * C(z) is analytic for z∈ℂ∖Γ_C, where Γ_C is shown in Figure <ref>.


  * C(z) satisfies the jump relation

    C_+(z)=C_-(z){   e^|x|^5/4g(z)σ_3S_ke^-|x|^5/4g(z)σ_3,       z∈Γ_k, k=±1,
       e^|x|^5/4g(z)σ_3S_ke^-|x|^5/4g(z)σ_3,       z∈Γ_k, k=±2,
       -iσ_1,       z∈(-∞,-α),
       e^|x|^5/4g_-(z)σ_3S_2(-iσ_1)S_-2
    e^-|x|^5/4g_+(z)σ_3,       z∈(-α,0),
       e^|x|^5/4g_-(z)σ_3S_0e^-|x|^5/4g_+(z)σ_3,       z∈(0,2α),
       e^|x|^5/4g(z)σ_3S_0e^-|x|^5/4g(z)σ_3,       z∈(2α,+∞).
    .



  * As z→∞, we have

    C(z)=[I+B_1/z+O(z^-2)]z^σ_3/41/√(2)[  1  1;  1 -1 ],

 where B_1 is given in (<ref>).






On the other hand, since g_±(z) is purely imaginary on (-∞,α), the diagonal entries of jumps on (-α,2α) are highly oscillating for large |x|. To turn the oscillations into exponential decays, we open lens around the segment (-α,2α) according to the sign of g(z) depicted in Figure <ref>. This process depends on the following matrix decomposition:

    [S_2(-iσ_1)S_-2]^-1=[           1 -is_2s_0^-1;           0           1 ][       0 -s_0^-1;     s_0       0 ][            1 -is_-2s_0^-1;            0            1 ],
       S_0=[      1 s_0^-1;      0      1 ][       0 -s_0^-1;     s_0       0 ][      1 s_0^-1;      0      1 ].

Owing to (<ref>) and (<ref>), we define

    D(z)=C(z){   [                     1 -s_0^-1e^2|x|^5/4g(z);                     0                     1 ],       z∈ℒ_1+,
       [                    1 s_0^-1e^2|x|^5/4g(z);                    0                    1 ],       z∈ℒ_1-,
       [                         1 is_-2s_0^-1e^2|x|^5/4g(z);                         0                         1 ],       z∈ℒ_2+,
       [                         1 -is_2s_0^-1e^2|x|^5/4g(z);                         0                         1 ],       z∈ℒ_2-,
    .

where the regions ℒ_1±, ℒ_2± are depicted in Figure <ref>. Therefore, we arrive at the following RH problem for D(z).


  * D(z) is analytic for z∈ℂ∖Γ_D, where Γ_D is shown in Figure <ref>.


  * D(z) satisfies the jump relations

    D_+(z)=D_-(z){   [                          1 -is_∓2s_0^-1e^2|x|^5/4g(z);                          0                          1 ],       z∈Γ_±1,
       [                   1                   0; s_±2e^-2|x|^5/4g(z)                   1 ],       z∈Γ_±2,
       [                     1 -s_0^-1e^2|x|^5/4g(z);                     0                     1 ],       z∈Γ^+_1∪Γ^-_1,
       [                          1 -is_∓2s_0^-1e^2|x|^5/4g(z);                          0                          1 ],       z∈Γ^±_2,
    .

and

    D_+(z)=D_-(z){   [  0 -i; -i  0 ],       z∈(-∞,-α),
       [       0 -s_0^-1;     s_0       0 ],       z∈(-α,2α),
       [                  1                  0; s_0e^-2|x|^5/4g(z)                  1 ],       z∈(2α,+∞).
    .



  * As z→∞, we have
  
    D(z)=[I+B_1/z+O(z^-2)]z^σ_3/41/√(2)[  1  1;  1 -1 ],

 where B_1 is given in (<ref>).






Again, it is readily seen from the signs of g(z) shown in Figure <ref> that as x→-∞, the jumps of D(z) tend to identity matrix except on (-∞,2α).








 §.§ Global parametrix

In this section, we construct the global parametrix. By (<ref>) and (<ref>), we consider the following model RH problem for a function N(z).


  * N(z) is analytic for z∈ℂ∖(-∞,2α), where (-∞,2α) is oriented from left to right.


  * N(z) satisfies the jump relation

    N_+(z)=N_-(z){   [ 0 i; i 0 ],       z∈(-∞,-α),
       [       0 -s_0^-1;     s_0       0 ],       z∈(-α,2α).
    .



  * As z→∞, we have
  
    N(z)=[I+O(z^-1)]z^σ_3/41/√(2)[  1  1;  1 -1 ].



It is straightforward to verify that

    N(z)=[         1 2√(3α) iν;         0         1 ]
    (z-2α)^σ_3/41/√(2)[  1  1;  1 -1 ]d(z)^σ_3,

where d(z) is the Szegő function

    d(z)   =exp(-ln(-is_0)/2π√(z-2α)∫^2α_-αdξ/√(2α-ξ) (ξ-z))
       =(i√(z-2α)+√(3α)/i√(z-2α)-√(3α))^ν

with

    ν=-1/2π iln(-is_0)=-1/2π iln(1-|s_2|^2)∈ iℝ.

The branches of the multi-valued functions in (<ref>) and (<ref>) are fixed by the conditions

    (z-2α)∈(-π,π),     (i√(z-2α)+√(3α)/i√(z-2α)-√(3α))∈(-π,π).

A direct calculation gives

    N(z)=[I+N_1/z+O(z^-2)]
    z^σ_3/41/√(2)[  1  1;  1 -1 ],     z→∞,

where

    N_1=[            6αν^2-α/2 2√(3)α^3/2iν(1-4ν^2);           -2√(3α) iν            α/2-6αν^2 ].









 §.§ Local parametrices

In this section, we construct relevant local parametrices in the neighborhood U(z_0)={z∈ℂ:|z-z_0|<δ} of the special point z_0∈{-α,0,2α}.

Firstly, we focus on the saddle point z_0=-α. In view of the RH problem for D(z), we consider the following local RH problem.


  * P^(-)(z) is analytic for z∈ U(-α)∖Γ_D.


  * P^(-)(z) satisfies the same jump relation as D(z) on Γ_D∩ U(-α).


  * For z∈∂ U(-α), we have
  
    P^(-)(z)N(z)^-1=I+O(|x|^-5/8),     x→-∞.




Let us define a conformal mapping

    λ_1(z)=2{   √(g_+(-α)-g(z)),          z>0,
       √(g_+(-α)+g(z)),          z<0,
    .

where the branch of the square roots are chosen by the condition

    λ_1(z)=e^-π i/42(3α)^1/4(z+α)[1-z+α/18α
    +O((z+α)^2)],     z→-α,

and

    g_+(-α)=-4/5√(3α) i.

Then, we construct P^(-)(z) as

    P^(-)(z)=E^(-)(z)Φ^(PC)(|x|^5/8λ_1(z))
    s_2^-σ_3/2h_1^σ_3/2M(z)e^-|x|^5/4g(z)σ_3,

where Φ^(PC) is the parabolic cylinder parametrix model with the parameter ν given in (<ref>) (see Appendix <ref>), h_1 is defined in (<ref>) and

    E^(-)(z)   =N(z)M(z)^-1h_1^-σ_3/2s_2^σ_3/2
    e^-|x|^5/4g_+(-α)σ_3(|x|^5/8λ_1(z))^νσ_32^-σ_3/2[ λ_1(z)      1;      1      0 ]

with

    M(z)={   iσ_1,       z>0,
       I,       z<0..

With (<ref>) and (<ref>), we check directly that P^(-)(z) fulfills the same jump relation as D(z) on Γ_D∩ U(-α) and E^(-)(z) is analytic in U(-α). In addition, we obtain from (<ref>), (<ref>) and (<ref>) that as x→-∞,

    P^(-)(z)N(z)^-1   =N(z)M(z)^-1[I+L(z)/|x|^5/8
    +O(|x|^-5/4)]M(z)N(z)^-1
       =I+O(|x|^-5/8),

where

    L(z)=[                                             0 ν s_2|x|^5/4νλ_1(z)^2ν-1/h_1e^2|x|^5/4g_+(-α);   h_1e^2|x|^5/4g_+(-α)/s_2|x|^5/4νλ_1(z)^2ν+1                                             0 ].

Notice that we have made use of the fact that ν and g_+(-α) are purely imaginary (see (<ref>) and (<ref>)).



For the local parametrix near the origin z_0=0, we consider the following local RH problem for a function P^(0)(z).


  * P^(0)(z) is analytic for z∈ U(0)∖Γ_D.


  * P^(0)(z) satisfies the same jump relation as D(z) on Γ_D∩ U(0).


  * For z∈∂ U(0), we have
  
    P^(0)(z)N(z)^-1=I+O(|x|^-∞),     x→-∞.



The solution to above problem is elementary. First, we note from the real solution condition (<ref>) that

    s_2=re^iβ,     s_-2=-re^-iβ,

where r:=|s_2|, β:= s_2. Define

    F(λ)=e^λσ_3{   I,      λ∈(π/4,3π/4)
    ∪(-3π/4,-π/4),
       [  1 ir;  0  1 ],       λ∈(3π/4,5π/4),
       [   1   0; -ir   1 ],       λ∈(-π/4,π/4).
    .

Then, we see that F(λ) solves the following RH problem.


  * F(λ) is analytic for λ∈ℂ∖{λ:λ=±π/4,±3π/4}, where rays are oriented from 0 to ∞.


  * F(λ) satisfies the jump relation

    F_+(λ)=F_-(λ){   [  1  0; ir  1 ],       λ=π/4,
       [  1 ir;  0  1 ],       λ=3π/4,
       [   1 -ir;   0   1 ],       λ=-3π/4,
       [   1   0; -ir   0 ],       λ=-π/4.
    .



  * As λ→∞, we have
  
    F(λ)=[I+O(λ^-∞)]e^λσ_3.



Introduce the conformal mapping

    λ_2(z)   =g_-(0)± g(z),    ± z>0,
       =e^π i/22√(2α)z(1+o(1)),     z→0.

We construct P^(0)(z) as

    P^(0)(z)=E^(0)(z)F(|x|^5/4λ_2(z))
    e^iβσ_3/2Q(z)
    s_0^σ_3/2e^-|x|^5/4g(z)σ_3,

where

    E^(0)(z)=N(z)s_0^-σ_3/2Q(z)^-1e^-iβσ_3/2
    e^-|x|^5/4g_-(0)σ_3

and

    Q(z)={   I,        z>0,
       [  0  1; -1  0 ],        z<0.
    .

Using (<ref>), (<ref>) and (<ref>), it is direct to see from (<ref>)-(<ref>) that P^(0)(z) satisfies the same jump relation as D(z) on Γ_D∩ U(0) and E^(0)(z) is analytic in U(0). Moreover, it follows from (<ref>) and the fact that g_-(0) is purely imaginary that the matching condition (<ref>) is also satisfied.


Finally, we concentrate on the branch point z_0=2α and we consider the following local RH problem for some function P^(+)(z).


  * P^(+)(z) is analytic for z∈ U(2α)∖Γ_D.


  * P^(+)(z) satisfies the same jump relation as D(z) on Γ_D∩ U(2α).


  * For z∈∂ U(2α), we have
  
    P^(+)(z)N(z)^-1=I+O(|x|^-5/4),     x→-∞.




As usual, in the neighborhood U(2α), we define a conformal mapping

    λ_3(z)=(3/2 g(z))^2/3=(6α)^2/3(z-2α)(1+o(1)),     z→2α.

Then the desired parametrix near z_0=2α is defined by

    P^(+)(z)=E^(+)(z)Φ^(Ai)(|x|^5/6λ_3(z))σ_1
    s_0^σ_3/2e^-|x|^5/4g(z)σ_3,

where Φ^(Ai) is the Airy parametrix model (see Appendix <ref>) and

    E^(+)(z)=N(z)s_0^-σ_3/2σ_11/√(2)[  1 -i; -i  1 ](|x|^5/6λ_3(z))^σ_3/4.







 §.§ Small norm problem

Using the model functions N(z) given in (<ref>), P^(-)(z) given in (<ref>), P^(0)(z) given in (<ref>) and P^(+)(z) given in (<ref>), we define the final transformation as

    R(z)=D(z){   P^(-)(z)^-1,       z∈ U(-α),
       P^(0)(z)^-1,       z∈ U(0),
       P^(+)(z)^-1,       z∈ U(2α),
       N(z)^-1,       elsewhere.
    .

As a result, we have the following RH problem for R(z).


  * R(z) is analytic for z∈ℂ∖Γ_R, where Γ_R is illustrated in Figure <ref>.


  * R(z) satisfies the jump relation R_+(z)=R_-(z)J_R(z)

    J_R(z)={   P^(-)(z)N(z)^-1,       z∈∂ U(-α),
       P^(0)(z)N(z)^-1,       z∈∂ U(0),
       P^(+)(z)N(z)^-1,       z∈∂ U(2α),
       N(z)J_D(z)N(z)^-1,       z∈Γ_R∖(∂ U(-α)∪∂ U(0)∪∂ U(2α)),
    .

where J_D(z) denotes the relevant matrix in (<ref>) and (<ref>).


  * As z→∞, we have
  
    R(z)=I+R_1/z+O(z^-2).







From the matching conditions (<ref>), (<ref>) and (<ref>), we see from (<ref>) that the jump matrix J_R(z) satisfy the estimation

    J_R(z)=I+O(|x|^-5/8),     x→-∞.

With the standard small norm theory <cit.>, we have

    R(z)=I+O(|x|^-5/8),     x→-∞,

uniformly for z∈ℂ∖Γ_R.








 §.§ The reduced case |s_2|=0

In the reduced case |s_2|=0, we obtain from (<ref>) that s_0=i. One sees from the RH problem for D(z) in Section <ref> that there are no jumps on Γ_±2∪Γ_2^±∪Γ_±1. The subsequent asymptotic analysis is almost same as the case 0<|s_2|<1.
First of all, we solve a RH problem with jump matrix iσ_1 on (-∞,2α). The solution to this RH problem can be constructed as (<ref>) by taking ν=0 therein. Next, we need to build a local parametrix near z=2α. The desired parametrix is defined as (<ref>) by taking |s_2|=0 therein. Similar to (<ref>), by defining R(z) as the ratio of the RH problem for D(z) and the parametrices, it is readily seen that R(z) satisfies the estimation R(z)=I+O(|x|^-5/4) as x→-∞.








§ ASYMPTOTICS OF Ψ FOR |S_2|>1

In this section, we will perform the Deift-Zhou nonlinear steepest descent analysis to the RH problem for Ψ(z,x) with |s_2|>1. As in the case 0<ρ<1, it includes a series of invertible transformations, and the first four transformations Ψ(z,x)→A(z)→B(z)→C(z)→D(z) are exactly the same as  those defined in  (<ref>), (<ref>), (<ref>) and(<ref>), respectively. The main differences lie in the constructions of the global and local parametrices. Indeed, by ignoring the exponentially small terms of J_D(z), we are led to the same RH problem for N(z). Since |s_2|>1, a solution is given by (<ref>) but with ν therein replaced by

    ν=-1/2-1/2πiln(|s_2|^2-1)
    :=-1/2+ν_0.

It is obvious that ν_0∈iℝ. If we continue to construct the local parametrix P^(-) as in (<ref>) and (<ref>), it comes out that P^(-)(z)N(z)^-1 does not tend to the identity matrix as t→+∞ for z∈∂ U(-α); see (<ref>) with ν given by (<ref>). This implies that the matching condition (<ref>) is no longer valid. To overcome this, our idea is to make some modifications of the global and local parametrices.





 §.§ Global parametrix

We modify the global parametrix by imposing specific singularity at z_0=-α. More precisely, we consider the following RH problem.


  * N(z) is analytic for z∈ℂ∖(-∞,2α), where (-∞,2α) is oriented from left to right.


  * N(z) satisfies the same jump relation as N(z); see (<ref>).


  * As z→∞, we have
  
    N(z)=[I+O(z^-1)]z^σ_3/41/√(2)[  1  1;  1 -1 ].



  * As z→-α, N(z) behaves like

    N(z)=O((z+α)^-3/2).



We look for a solution in the form

    N(z)=(X/z+α+I)N(z),    (X/z+α+I)=1,

where N(z) is given via (<ref>) and (<ref>). The z-independent matrix X is to be determined. It is easy to see that

    N(z)=[I+X+N_1/z+O(z^-2)]
    z^σ_3/41/√(2)[  1  1;  1 -1 ],     z→∞,

where N_1 is given in (<ref>).






 §.§ Local parametrices

Near the saddle point z_0=-α, we instead consider the following local RH problem.


  * P^(-)(z) is analytic for z∈ U(-α)∖Γ_D.


  * P^(-)(z) satisfies the same jump relation as D(z) on Γ_D∩ U(-α).


  * For z∈∂ U(-α), we have
  
    P^(-)(z)N(z)^-1=I+O(|x|^-5/4),     x→-∞.



We define the local parametrix P^(-)(z) as

    P^(-)(z)=E^(-)(z)
    Φ^(PC)(|x|^5/8λ_1(z))
    s_2^-σ_3/2h_1^σ_3/2M(z)e^-|x|^5/4g(z)σ_3,

where Φ^(PC) is the parabolic cylinder parametrix model with the parameter ν given in (<ref>) (see Appendix <ref>), λ_1(z) is the conformal mapping (<ref>) and

    E^(-)(z)=N(z)M(z)^-1h_1^-σ_3/2   s_2^σ_3/2e^-|x|^5/4g_+(-α)σ_3(|x|^5/8λ_1(z))^νσ_3
       ×[                1                0; -1/|x|^5/8λ_1(z)                1 ]2^-σ_3/2[ λ_1(z)      1;      1      0 ].


Using the jumps (<ref>), (<ref>) and (<ref>), we see that P^(-)(z) has the same jump matrices as D(z) on Γ_D∩ U(-α) and E^(-)(z) is analytic in U(-α)∖{-α}. In addition, with the asymptotic behavior (<ref>), it is easily seen that the matching condition (<ref>) is fulfilled. To make sure that E^(-)(z) is also analytic at the isolated point z=-α, we have to make an appropriate choice of X in (<ref>). Indeed, by calculating the Laurent expansion at z=-α of E^(-)(z) using (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have

    E^(-)(z)=(X/z+α+I)
    (H/z+α+I)Y(z),

where Y(z) is analytic near z_0=-α and

    H=6α c/1-c[          2ν_0 -4√(3α)iν_0^2;      -i/√(3α)         -2ν_0 ],

with

    c=c(x)=h_1e^2|x|^5/4g_+(-α)/s_2|x|^5/4ν_0e^3π iν_0/22^6ν_0(3α)^5ν_0/2.

Since H^2=0, it follows from (<ref>) that the X is determined by

    X=-H=6α c/1-c[        -2ν_0 4√(3α)iν_0^2;      i/√(3α)         2ν_0 ].

With this choice of X, it is clear that (X/z+α+I)=1. It should be mentioned that we assume in (<ref>) and (<ref>) that x does not belong to the zero set of the function 1-c(x) (see (<ref>) below).

Since X/z+α+I is analytic at z=0,2α, the parametrices near these points can be just defined by replacing N(z) by N(z) in (<ref>), (<ref>) and (<ref>) and (<ref>).






 §.§ Small norm problem

With the modified model function N(z) in (<ref>), P^(-)(z) in (<ref>), P^(0)(z) in (<ref>) and P^(+)(z) in (<ref>), the final transformation is defined by

    R(z)=D(z){   P^(-)(z)^-1,       z∈ U(-α),
       P^(0)(z)^-1,       z∈ U(0),
       P^(+)(z)^-1,       z∈ U(2α),
       N(z)^-1,       elsewhere.
    .

Therefore, we get the following RH problem for R(z).


  * R(z) is analytic for z∈ℂ∖Γ_R, where Γ_R is shown in Figure <ref>.


  * R(z) satisfies the jump relation R_+(z)=R_-(z)J_R(z)

    J_R(z)={   P^(-)(z)N(z)^-1,       z∈∂ U(-α),
       P^(0)(z)N(z)^-1,       z∈∂ U(0),
       P^(+)(z)N(z)^-1,       z∈∂ U(2α),
       N(z)J_D(z)N(z)^-1,       z∈Γ_R∖(∂ U(-α)∪∂ U(0)∪∂ U(2α)),
    .

where J_D(z) denotes the jump matrix in (<ref>) and (<ref>).


  * As z→∞, we have
  
    R(z)=I+R_1/z+O(z^-2).





It is readily seen from (<ref>), (<ref>) and (<ref>) that

    J_R(z)=I+O(|x|^-5/4),      x→-∞.

Thus, we have that for z∈ℂ∖Γ_R,

    R(z)=I+O(|x|^-5/4),      x→-∞.









§ PROOF OF THEOREM <REF>

In this section, we prove  Theorem <ref>. The proof depends on the formulas (<ref>) and the asymptotic analysis of the RH problem for Ψ(z,x) given in the previous two sections. Below, it should be understood that |s_2|<1 in Section <ref> and |s_2|>1 in Section <ref>.



 §.§ Derivation of (<ref>) and (<ref>)

By tracing back the transformations Ψ(z,x)→ A(z)→ B(z)→ C(z)→ D(z)→ R(z) as given in (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we obtain that for z∈ℂ∖Γ_R,

    D(z)=R(z)N(z).

Taking the limit z→∞ in (<ref>), it follows from the expansions (<ref>), (<ref>) and (<ref>) that

    B_1,21   =R_1,21+N_1,21,
    
    B_1,22   =R_1,22+N_1,22.

In views of (<ref>)-(<ref>), we obtain from (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) that

    y(x)   =-(α+4√(3α) iν R_1,21+2R_1,22-R_1,21^2)|x|^1/2,
    ℋ(x)   =-2α/3|x|^3/2
    +2√(3α) iν|x|^1/4-R_1,21|x|^1/4.


Next, we compute R_1. Since the RH problem for R(z) is equivalent to the following singular integral equation

    R(z)=I+1/2π i∫_Γ_RR_-(ξ)(J_R(ξ)-I)dξ/ξ-z,

we get from (<ref>), (<ref>) and (<ref>) that as x→-∞,

    R_1   =-1/2π i∫_Γ_RR_-(ξ)(J_R(ξ)-I)dξ
       =-1/2π i∫_∂ U(-α)(J_R(ξ)-I)dξ+O(|x|^-5/4)
       =-|x|^-5/8/2π i∫_∂ U(-α)N(ξ)M(ξ)L(ξ)M(ξ)^-1N(ξ)^-1dξ
    +O(|x|^-5/4).

Via a direct residue calculation using (<ref>), (<ref>), (<ref>) and (<ref>), we obtain that as x→-∞,

    R_1,21   =i/2√(3α)(A-B)|x|^-5/8+O(|x|^-5/4),
    
    R_1,22   =[-1/2(A+B)+ν(A-B)]|x|^-5/8+O(|x|^-5/4),

where

    A=ν s_2|x|^5/4νe^3π iν/22^6ν(3α)^5ν/2/h_1e^2|x|^5/4g_+(-α)
    e^-π i/42(3α)^1/4,    
    B=h_1e^2|x|^5/4g_+(-α)/s_2|x|^5/4νe^3π iν/22^6ν(3α)^5ν/2e^-π i/42(3α)^1/4.

Inserting the asymptotics in (<ref>), (<ref>) into (<ref>) and (<ref>) gives

    y(x)   =-α|x|^1/2+(A+B)|x|^-1/8+O(|x|^-3/4),
    ℋ(x)   =-2α/3|x|^3/2
    +2√(3α) iν|x|^1/4-i(A-B)/2√(3α)|x|^-3/8
    +O(|x|^-1),

as x→-∞.

Using (<ref>) and (<ref>), we immediately have

    A± B=νΓ(-ν) s_2|x|^5/4νe^π iν/22^6ν(3α)^5ν/2/√(2π)e^2|x|^5/4g_+(-α)
    e^-π i/42(3α)^1/4±√(2π)h_1e^2|x|^5/4g_+(-α)/Γ(-ν)s_2|x|^5/4νe^π iν/22^6ν(3α)^5ν/2e^-π i/42(3α)^1/4.

With (<ref>), (<ref>), the relation Γ(-ν)=Γ(ν) and the reflection formula of the Gamma function

    -νΓ(-ν)Γ(ν)=π/sinπν=2π i/e^π iν|s_2|^2,

we find that

    A+B   =√(2π)/(3α)^1/4|s_2||Γ(ν)|e^π iν/2cosφ(x),
    
    A-B   =i√(2π)/(3α)^1/4|s_2||Γ(ν)|e^π iν/2sinφ(x),

where

    φ(x)=2ig_+(-α)|x|^5/4-iνln[2^6(3α)^5/2|x|^5/4]-Γ(ν)+ s_2-π/4.

On the other hand, it follows from (<ref>) that

    |Γ(ν)|=√(2π)/√(iν) e^π iν/2|s_2|.

Substituting (<ref>)-(<ref>) into (<ref>) and (<ref>) and then recalling (<ref>), (<ref>) and (<ref>), we arrive at (<ref>) and (<ref>) for the case 0<|s_2|<1.

In the reduced case |s_2|=0, by applying the asymptotic analysis of the RH problem  for Ψ discussed in Section <ref>, we obtain the following asymptotic formula

    y(x)   =-(-x/6)^1/2+O(x^-3/4),
    ℋ(x)   =-4(-x/6)^3/2+O(x^-1),

as x→-∞. We notice from (<ref>) that ν=0 when |s_2|=0. Thus, (<ref>) and (<ref>) can be viewed respectively as the limits of (<ref>) and (<ref>) as a→0.



Note that both the parabolic cylinder parametrix and Airy parametrix have a full asymptotic expansion as their variables tend to infinity. Hence, the jump matrices of R(z) on ∂ U(-α) and ∂ U(2α) can also be fully asymptotically expanded as |x|→+∞. This indicates that we can theoretically obtain the full asymptotic expansion of the real solutions of Painlevé I equation. We conjecture that (<ref>) can be improved as

    y(x)∼√(-x/6)∑_k=0^∞a_k/(-x)^5k/4
    +√(a)/(-24x)^1/8∑_n=1^∞cos(nψ(x))
    ∑_k=0^∞b_k^(n)/(-x)^5k/8,

where ψ(x)=4·24^1/4/5(-x)^5/4-5a/8ln(-x)+ϕ and a_k, b_k^(n) are constants. One may try to determine the coefficients a_k and b_k^(n) by following the way above in this section, while we think it is better to obtain these coefficients by substituting (<ref>) into the Painlevé I equation (<ref>) directly.







 §.§ Derivation of (<ref>) and (<ref>)

Inverting the transformations Ψ(z,x)→ A(z)→ B(z)→ C(z)→ D(z)→R(z) as given in (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we now get that for z∈ℂ∖Γ_R,

    D(z)=R(z)N(z).

Letting z→∞ in (<ref>), we obtain from the expansions (<ref>), (<ref>) and (<ref>) that

    B_1,21   =R_1,21+N_1,21+X_21,
    
    B_1,22   =R_1,22+N_1,22+X_22.

Using (<ref>)-(<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have

    y(x)   =-α|x|^1/2-12α c/(1-c)^2|x|^1/2-2R_1,22|x|^1/2-R_1,21^2|x|^1/2
                -4√(3α) iν_0 R_1,21|x|^1/2
    +2√(3α)iR_1,211-c/1+c|x|^1/2,
    ℋ(x)   =-2α/3|x|^3/2
    +2√(3α) iν_0|x|^1/4-i√(3α)(1+c)/1-c|x|^1/4
    -R_1,21|x|^1/4.


For the asymptotics of R_1, we use the singular integral equation for R(z) and the estimations (<ref>), (<ref>) to deduce that
as x→-∞,

    R_1=-1/2π i∫_Γ_RR_-(ξ)
    (J_R(ξ)-I)dξ=O(|x|^-5/4).

On the other hand, it follows from (<ref>) and (<ref>) that

    c=√(2π)e^-1/2π ie^2|x|^5/4g_+(-α)/Γ(1/2-ν_0)s_2|x|^5/4ν_0e^π iν_0/22^6ν_0(3α)^5ν_0/2.

Since

    |Γ(1/2-ν_0)|^2=Γ(1/2-ν_0)Γ(1/2+ν_0)
    =π/sinπ(1/2-ν_0)=2π/e^π iν_0|s_2|^2,

we have

    c=exp{i(-2ig_+(-α)|x|^5/4+5/4iν_0ln|x|
    .   +iν_0ln[2^6(3α)^5/2]
       .- s_2-Γ(1/2-ν_0)-π/2)}.

Hence, by (<ref>),

    c/(1-c)^2=-1/4sin^2ω(x),      1+c/1-c=-iω(x),

where

    ω(x)=ig_+(-α)|x|^5/4-5/8iν_0ln|x|   -
    1/2iν_0ln[2^6(3α)^5/2]
       +1/2 s_2+1/2Γ(1/2-ν_0)+π/4.

Plugging the asymptotics (<ref>), the equations (<ref>) and (<ref>) into (<ref>), (<ref>) and then recalling (<ref>), (<ref>) and (<ref>), we obtain (<ref>) and (<ref>).








§ ACKNOWLEDGEMENTS

The work of Jun Xia was supported in part by the National Natural Science Foundation of China [Grant no. 12401322] and the Guangdong Basic and Applied Basic Research Foundation (Grant no. 2024A1515012985).
The work of Wen-Gao Long was supported in part by the National Natural Science Foundation of China
[Grant no. 12401094], the Natural Science Foundation of Hunan Province [Grant no. 2024JJ5131] and the Outstanding Youth Fund of Hunan Provincial Department of Education [Grant no. 23B0454].








§ MODEL PARAMETRICES



 §.§ Parabolic cylinder  parametrix

Let

    𝐏(λ)=2^-σ_3/2[  D_-ν-1(iλ)      D_ν(λ); D_-ν-1'(iλ)     D_ν'(λ) ][ e^i π/2(ν+1)            0;            0            1 ],

where D_ν(λ) is parabolic cylinder function (see <cit.>).
Define

    H_0=[   1   0; h_0   1 ], H_1=[   1 h_1;   0   1 ], H_n+2=e^i π(ν+1/2) σ_3 H_n e^-i π(ν+1/2) σ_3,  n=0,1,

where

    h_0=-i √(2 π)/Γ(ν+1),    h_1=√(2 π)/Γ(-ν) e^i πν,    1+h_0 h_1=e^2 π i ν.

We introduce

    Φ^(PC)(λ)={   𝐏(λ),      λ∈(-π/4, 0), 
       𝐏(λ)H_0,      λ∈(0,π/2), 
       𝐏(λ)H_0H_1,      λ∈(π/2, π), 
       𝐏(λ)H_0H_1H_2,      λ∈(π,3π/2), 
       𝐏(λ)H_0H_1H_2H_3,      λ∈(3π/2,7π/4).
    .

Thus, Φ^(PC) solves the following RH problem (cf. <cit.>).


  * Φ^(PC) is analytic for λ∈ℂ∖⋃^5_k=1Σ_k, where Σ_k={λ∈ℂ:λ=(k-1)π/2}, k=1,2,3,4 and  Σ_5={λ∈ℂ:λ=-π/4}; see Figure <ref>.

  * Φ^(PC) satisfies the jump conditions

    Φ^(PC)_+(λ)=Φ^(PC)_-(λ){   H_0,       λ∈Σ_1,
       H_1,       λ∈Σ_2,
       H_2,       λ∈Σ_3,
       H_3,       λ∈Σ_4,
       e^2π iνσ_3,       λ∈Σ_5.
    .


  * As λ→∞, we have

    Φ^(PC)(λ)=[  0  1;  1 -λ ]2^σ_3/2[I+∑_k=1^∞Φ^(PC)_kλ^-k]
    e^(λ^2/4-νlnλ) σ_3,

where Φ^(PC)_2k-1 is off-diagonal and Φ^(PC)_2k is diagonal,

    Φ^(PC)_1=[ 0 ν; 1 0 ],    Φ^(PC)_2=[  ν(ν+1)/2         0;         0 -ν(ν-1)/2 ].
















 §.§ Airy parametrix

Let φ=e^2π i/3. We consider

    Φ^(Ai)(λ)=Υ{   [         Ai(λ)      Ai(φ^2λ);       Ai^'(λ) φ^2Ai^'(φ^2λ) ] e^-i π/6σ_3,       λ∈I,
       [         Ai(λ)      Ai(φ^2λ);       Ai^'(λ) φ^2Ai^'(φ^2λ) ] e^-i π/6σ_3[  1  0; -1  1 ],      λ∈II,
       [      Ai(λ) -φ^2Ai(φλ);    Ai^'(λ)  -Ai^'(φλ) ] e^-i π/6σ_3[ 1 0; 1 1 ],      λ∈III,
       [      Ai(λ) -φ^2Ai(φλ);    Ai^'(λ)  -Ai^'(φλ) ] e^-i π/6σ_3,      λ∈IV,
    .

where Ai(λ) is the Airy function (cf. <cit.>) and

    Υ=√(2π)[  e^1/6π i         0;         0 e^-1/3π i ].

It is direct to see that Φ^(Ai) satisfies the following RH problem (cf. <cit.>).


  * Φ^(Ai) is analytic for λ∈ℂ∖⋃^4_k=1Σ_k, where Σ_k is depicted in Figure <ref>.

  * Φ^(Ai) satisfies the jump conditions

    Φ^(Ai)_+(λ)=Φ^(Ai)_-(λ){   [ 1 1; 0 1 ],    λ∈Σ_1,
       [ 1 0; 1 1 ],    λ∈Σ_2∪Σ_4,
       [  0  1; -1  0 ],    λ∈Σ_3.
    .


  * Φ^(Ai) satisfies the asymptotic behavior

    Φ^(Ai)(λ)=λ^-σ_3/41/√(2)[ 1 i; i 1 ][I+O(λ^3/2)] e^-2/3λ^3/2σ_3,  λ→∞.







§ STOKES MULTIPLIERS OF A SPECIAL SOLUTION OF PAINLEVÉ I 

According to <cit.> (see also <cit.>), Ψ(z,x) satisfies the following differential equation

    ∂Ψ/∂ z
    =[             y_x 2z^2+2yz+x+2y^2;          2(z-y)            -y_x ]Ψ.

The only singularity of the above equation is the irregular singular point at z=∞.
Following <cit.> (see also <cit.>),
there exist the canonical solutions Ψ_k(z,x), k∈ℤ, of (<ref>) with the asymptotic expansion

    Ψ^(k)(z,x)
    =z^1/4σ_3σ_3+σ_1/√(2)[I+-ℋ(x)σ_3/√(z)
    +O(1/z)]
    e^(4/5λ^5/2+xz^1/2)σ_3

as z→∞ with z∈Ω_k, uniformly for all x bounded away from the poles of the Painlevé I transcendents, where ℋ(x) is introduced in (<ref>), and the canonical sectors are

    Ω_k={z∈ℂ:  z∈(-3π/5+2kπ/5,π/5+2kπ/5)},      k∈ℤ.

These canonical solutions are connected by

    Ψ^(k+1)(z,x)=Ψ^(k)(z,x)S_k,   S_2k-1=[      1 s_2k-1;      0      1 ],   S_2k=[    1    0; s_2k    1 ].

By <cit.>, we can proceed further as follows.
Define

    Ψ̂(z,x)=[                   0                   1;                   1 -1/2(-y_x+1/2(z-y)) ]
    (z-y)^σ_3/2Ψ(z,x)

Then Φ̂(z,x) satisfies

    ∂/∂ zΨ̂(z,x)=[      0      2; V(z,x)      0 ]Ψ̂(z,x),

where

    2V(z,x)=y_x^2+4z^3+2z x-2y x-4y^3-y_x/z-y+3/41/(z-y)^2.

Moreover, using the expansion of Ψ(z,x) in (<ref>), one can verify that

    Ψ̂^(k)(z,x)
    =z^-3/4σ_3/√(2)[  1 -1;  1  1 ](I+O(z^-1/2))
    e^(4/5z^5/2+xz^1/2)σ_3

as z→∞ with z∈Ω_k.
The asymptotic expansions of Ψ^(k)(z,x) and Ψ̂^(k)(z,x) in (<ref>) and (<ref>) are valid
only when x is bounded away from p.
When x→ p, according to <cit.>,
the system (<ref>) reduces to

    ∂/∂ zΨ̂(z,p)
    =[           0           2; 2z^3+pz-14H           0 ]Ψ̂(z,p),

and the corresponding asymptotic expansions of Ψ̂_k(z,p) in (<ref>) should be replaced by

    Ψ̂^(k)(z,p)
    =z^-3/4σ_3/√(2)[ -i -i; -i  i ](I+O(z^-1/2))
    e^(4/5z^5/2+pz^1/2)σ_3

as z→∞ with z∈Ω_k; see <cit.>. It is readily seen that Ψ̂^(k) also satisfy the connection formula

    Ψ̂^(k+1)(z,p)=Ψ̂^(k)(z,p)S_k, k∈ℤ.

Finally, if denoting

    Ψ̂(z,p)=[ ϕ_1; ϕ_2 ],

and letting Y(z;p)=ϕ_1, then we arrive at the following reduced triconfluent Heun equation (RTHE); see  <cit.>

    d^2Y/d z^2=[4z^3+2pz-28H]Y.


When (p,H)=(0,0), the equation (<ref>) can be explicitly solved by the Bessel functions. For any solution of (<ref>), there exist two constants C_1 and C_1 such that

    Y(z)=C_1z^1/2I_1/5(4/5z^5/2)
    +C_2z^1/2K_1/5(4/5z^5/2).

Applying the asymptotic behaivor of the Bessel functions I_ν(z) and K_ν(z) (cf. see <cit.>) and matching them with the behaviors of (Ψ̂^(k))_11 and (Ψ̂^(k))_12 in (<ref>), we have

    (z^1/2I_1/5(4/5z^5/2),
    z^1/2K_1/5(4/5z^5/2))   ∼ ((Ψ^(0))_11,(Ψ^(0))_12)
    (
    c_1   0
    
    c_2   c_3)

as z→∞ with z∼ -π/5, where

    c_1=(4π/5)^-1/2i,   c_2=(4π/5)^-1/2e^-π i/5,   c_3=(4/5π)^-1/2i.

Similarly when z→∞ with z∼π/5, we have

    (z^1/2I_1/5(4/5z^5/2),
    z^1/2K_1/5(4/5z^5/2))   ∼ ((Ψ^(1))_11,(Ψ^(1))_12)
    (
    c_1   0
    c̃_2   c_3),

where c̃_2=-e^2π i/5c_2=ie^π i/5c_1.
Hence, according to the connection formula (<ref>), we conclude that

    S_0=(
    c_1   0
    
    c_2   c_3)(
    c_1   0
    c̃_2   c_3)^-1
    =(
    1   0
    c_2-c̃_2/c_1   1
    ),

which implies that s_0=c_2-c̃_2/c_1=-2icos(π/5).

Combining <cit.> with the connection formulas between I_ν(z) and K_ν(z) (cf. see <cit.>), we can further obtain

    (z^1/2I_1/5(4/5z^5/2),
    z^1/2K_1/5(4/5z^5/2))
    ∼((Ψ^(2))_11,(Ψ^(2))_12)(
    d_1   d_3
    d_2   d_4),

as z→∞ with z∼3π/5,
where

    d_1=ie^π i/5c̃_2=-c_1,    d_2=ie^π i/5c_1=-c_2,   d_3=e^-π i/5ic_3+πc̃_2,   d_4=π c_1=c_3.

Comparing (<ref>) and (<ref>), it is readily seen that

    S_1=(
    c_1   0
    c̃_2   c_3)(
    d_1   d_3
    d_2   d_4)^-1=(
    1   -c_1d_3
    0   1
    ).

This implies that s_1=-c_1d_3=-2icos(π/5).

Finally, making use of (<ref>), we find that s_k=-2icos(π/5) for all k∈ℤ.










99

BO
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and
Engineers I, Springer Science & Business Media, New York, 1999.

Bertola-Tovbis-2013
M. Bertola and A. Tovbis,
Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point:
rational breathers and poles of the tritronquée solution to Painlevé I,
Comm. Pure Appl. Math., 66 (2013), pp. 678–752.

Deano
A. Deaño, On the Riemann-Hilbert approach to asymptotics
of tronquée solutions of Painlevé I, J. Phys. A, 56 (2023), 314001.

Deift
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, vol. 3, New York University, 1999.

DZ1
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math., 137 (1993), 295-368.

DZ2
P. Deift and X. Zhou,  Asymptotics for the Painlevé II equation, Commun. Pure Appl. Math., 48 (1995), 277-337.

FIKN
A. S. Fokas, A. R. Its, A. A. Kapaev and V.Yu. Novokshenov, Painlevé transcendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.

GLS
V. I. Gromak, I. Laine and S. Shimomura, Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter GmbH & Co. KG, Berlin, 2002.

HS
P. Holmes and D. Spence, On a Painlevé-type boundary-value problem, Quart. J.
Mech. Appl. Math., 37 (1984), 525-538.

JK
N. Joshi and M. D. Kruskal, The Painlevé connection problem: An asymptotic
approach. I, Stud. Appl. Math., 86 (1992), 315-376.


Kap1
A. A. Kapaev, Asymptotic behavior of the solutions of the Painlevé equation of the first kind, Differ. Uravn. 24 (1988), 1684-1695.

Kap2
A. A. Kapaev, Quasi-linear Stokes phenomenon for the Painlevé
first equation, J. Phys. A, 37 (2004), 11149–11167.

Kapaev-Kitaev-1993
A. A. Kapaev and A. V. Kitaev, Connection formulae for the first Painlevé transcendent in the complex domain, Lett. Math. Phys., 27 (1993), 243–252.

Kitaev
A. V. Kitaev,
Symmetric solutions for the first and the second Painlevé equation, J. Math. Sci., 73 (1995),  494–499


LongLi
W.-G. Long and Y.-T. Li
Connection problem of the first Painlevé transcendents with large initial data,
J. Phys. A: Math. Theor., 56 (2023), 175201.

LongLLZ
W.-G. Long, Y.-T. Li, S.-Y. Liu and Y.-Q. Zhao,
Real solutions of the first Painlevé equation with large initial data,
Stud. Appl. Math., 139 (2017), 505–532.

LongLiWang
W.-G. Long, Y.-T. Li, and Q.-h. Wang,
Connection problem of the first Painlevé transcendent between poles and negative infinity,
SIAM J. Math. Anal., 55 (2023), 6676–6706.

NIST
F. W. J. Olver, A. B. O. Daalhuis, D. W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders (Eds.), 2020, NIST Digital Library of Mathematical Functions, Release 1.0.28 of 2020-09-15, http://dlmf.nist.gov/.

QL
H.-Z. Qin and Y.-M. Lu, A note on an open problem about the first Painlevé
equation, Acta Math. Appl. Sin. Engl. Ser., 24 (2008), 203-210.

Xia-Xu-Zhao
J. Xia, S.-X. Xu and Y.-Q. Zhao, Isomonodromy sets of accessory parameters for Heun class equations, 
Stud. Appl. Math., 146 (2021), 901–952.


