Asymptotics for the Sasa–Satsuma equation

Asymptotics for the Sasa–Satsuma equation
 in terms of a modified Painlevé II transcendent

Lin Huang School of sciences, Hangzhou Dianzi University, 310018, Hangzhou, China. lin.huang@hdu.edu.cn  and  Jonatan Lenells Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden. jlenells@kth.se
Abstract.

We consider the initial-value problem for the Sasa–Satsuma equation on the line with decaying initial data. Using a Riemann–Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector , constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painlevé II equation. Whereas the standard Painlevé II equation is related to a matrix Riemann–Hilbert problem, this modified Painlevé II equation is related to a matrix Riemann–Hilbert problem.

AMS Subject Classification (2010): 35Q15, 37K15, 41A60.

Keywords: Sasa–Satsuma equation, Riemann–Hilbert problem, asymptotics, initial value problem.

1. Introduction

In this paper, we consider the long-time behavior of the solution of the Sasa–Satsuma equation [11]

(1.1)

with initial data in the Schwartz class. Our main result shows that admits an expansion to all orders in the asymptotic sector of the form

(1.2)

where are smooth functions of and is a constant. It also shows that the leading coefficient is given by

where satisfies the following modified Painlevé II equation:

(1.3)

Equation (1.3) coincides with the standard Painlevé II equation

(1.4)

except for a sign difference and the presence of the absolute value squared in the last term. We will show that (1.3) is related to a matrix RH problem much in the same way that (1.4) is related to a matrix RH problem cf. [5]. In the case of a real-valued solution, equation (1.1) reduces to a version of the mKdV equation, (1.3) reduces (up to a sign) to (1.4), and the expansion (1.2) reduces to the analogous asymptotic formula for the corresponding mKdV equation (see [4], and [2] for the higher order terms, in the case of the standard mKdV equation).

It turns out that the leading coefficient in (1.2) has constant phase, that is, where is independent of . It is somewhat remarkable that this is the case for any choice of the complex-valued initial data ; however, we also recall that the Sasa–Satsuma has a class of one-soliton solutions of constant phase (see [11] or [1]):

The starting point for our analysis is a Riemann–Hilbert (RH) representation for the solution of (1.1) obtained via the inverse scattering transform formalism. The asymptotic formula (1.2) is derived by performing a Deift–Zhou [4] steepest descent analysis of this RH problem. The main novelty compared with the analogous derivation for the mKdV equation is that the Lax pair of (1.1) involves instead of matrices.

The inverse scattering problem for (1.1) was studied already by Sasa and Satsuma [11]. The initial-boundary value problem for (1.1) on the half-line was considered in [12]. Asymptotic formulas for the long-time behavior in the sector were obtained in [6, 10].

Our main results are presented in Section 2. They are stated in the form of three theorems (Theorem 1-3) whose proofs are given in Section 4, 5, and 6, respectively. Section 3 recalls the Lax pair formulation of (1.1). The RH problem associated with the modified Painlevé II equation (1.3) is discussed in Appendix A. Appendix B considers an extension of this RH problem which is needed to obtain the higher order terms in (1.2).

2. Main results

Our first theorem shows how solutions of (1.1) can be constructed starting from an appropriate spectral function . We let denote the Schwartz class of smooth (complex-valued) rapidly decaying functions.

Theorem 1 (Construction of solutions).

Suppose . Define the -matrix valued jump matrix by

(2.1)

where

Then the -matrix RH problem

  • is analytic for and extends continuously to from the upper and lower half-planes;

  • the boundary values obey the jump condition for ;

  • as ;

has a unique solution for each and the limit exists for each . Moreover, the function defined by

(2.2)

is a smooth function of with rapid decay as which satisfies the Sasa–Satsuma equation (1.1) for .

Proof.

See Section 4. ∎

Our second theorem gives the long-time asymptotics of the solutions constructed in Theorem 1 in the sector .

Theorem 2 (Asymptotics of constructed solutions).

Under the assumptions of Theorem 1, the solution of (1.1) defined in (2.2) satisfies the following asymptotic formula as :

(2.3)

where

  • The formula holds uniformly with respect to in the given range for any fixed and .

  • The variable is defined by

  • are smooth functions of .

  • The function is given by

    (2.4)

    where and denotes the smooth solution of the modified Painlevé II equation (1.3) corresponding to according to Lemma A.1. In particular, has a constant phase, that is, is independent of .

Proof.

See Section 5. ∎

Remark 2.1 (Hierarchy of differential equations).

Substituting the expansion (2.3) into (1.1) and identifying coefficients of powers of , we infer that the coefficients in (2.3) satisfy a hierarchy of linear ordinary differential equations. The first two equations in this hierarchy are

(2.5a)
(2.5b)

As expected, the function in (2.4) satisfies the first of these equations. Indeed, if is given by (2.4) where satisfies (1.3), then (2.5a) reduces to the equation , which is satisfied for solutions of constant phase.

By applying the above two theorems in the case when is the “reflection coefficient” corresponding to some given initial data , we obtain our third theorem, which establishes the asymptotic behavior of the solution of the initial-value problem for (1.1) in the sector . Before stating the theorem, we introduce some notation.

Given , define and by

Define the -matrix valued function as the unique solution of the Volterra integral equation

where acts on a matrix by , i.e., . Define the scattering matrix by

(2.6)

Then the “reflection coefficient” is defined by

(2.7)

We will see in Section 6 that the entry of has an analytic continuation to the upper half-plane. Possible zeros of give rise to poles in the RH problem, see (6.8). For simplicity, we assume that no such poles are present (solitonless case).

Theorem 3 (Asymptotics for initial value problem).

Suppose and define and by (2.6) and (2.7). Suppose the (33)-entry is nonzero for .

Then and the solution of (1.1) defined in terms of by (2.2) is the unique solution of the initial value problem for (1.1) with initial data and rapid decay as . Moreover, obeys the asymptotic formula (2.3) as .

Proof.

See Section 6. ∎

Remark 2.2 (Scattering transform).

Let denote the subset of consisting of all functions such that the associated scattering matrix defined in (2.6) satisfies for . Theorem 3 shows that the map which takes to (the scattering transform) is a bijection from onto its image in . The inverse of this map (the inverse scattering transform) is given by the construction of Theorem 1 for .

3. Lax pair

An essential ingredient in the proofs of Theorem 1-3 is the fact that equation (1.1) is the compatibility condition of the Lax pair equations [11]

(3.1)

where is the spectral parameter, is a -matrix valued eigenfunction, the -matrix valued functions and are defined by

(3.2)

where , ,

(3.3)
(3.4)

Note that and are rapidly decaying as if is, and that obey the symmetries

(3.5a)
(3.5b)

where denotes the complex conjugate transpose of a matrix   and

4. Proof of Theorem 1

Suppose . The associated jump matrix defined in (2.1) obeys the symmetries

(4.1)

In particular, is Hermitian and positive definite for each . Hence the result of Zhou [13] implies that there exists a vanishing lemma for the RH problem for , i.e., the associated homogeneous RH problem has only the zero solution.

Defining the nilpotent matrices by

we can write , where . For , we define the Cauchy transform by

(4.2)

and denote the nontangential boundary values of from the left and right sides of by and , respectively. Then and are bounded operators on and . Given two functions , we define the operator by

(4.3)

For each , we have and . In view of the vanishing lemma, this implies (see e.g. [9, Theorem 5.10]) that is an invertible bounded linear operator on , and that the matrix -RH problem for has a unique solution for each given by

where

The smoothness and decay of together with the smooth dependence on implies that is a classical solution of the RH problem and that admits an expansion

(4.4)

where the coefficients are smooth functions of (see e.g. [8, Section 4] for details in a similar situation). Since , an application of the Deift-Zhou steepest descent method [4] implies that and the coefficients have rapid decay as for each . In particular, the limit in (2.2) exists for each and is a smooth function of with rapid decay as .

Lemma 4.1.

Define by (2.2). Then

(4.5)

where and are defined in terms of by (3.3) and (3.4), respectively.

Proof.

The symmetries (4.1) of together with the uniqueness of the solution of the RH problem imply the following symmetries for :

(4.6)

In particular, the coefficient in (4.4) satisfies

It follows that the definition (2.2) of can be expressed as

(4.7)

Define the operator by

(4.8)

Substituting the expansion (4.4) into (4.8), we find

In view of (4.7), this implies that satisfies the following homogeneous RH problem:

  • is analytic in with continuous boundary values on ;

  • for ;

  • as .

Thus, by the vanishing lemma, . This proves the first equation in (4.5).

In order to prove the second equation in (4.5), we define the operator by

(4.9)

where the matrices and are yet to be determined. Substituting the asymptotic expansion (4.4) into (4.9), we find

Thus, we define by the equations

(4.10a)
(4.10b)
(4.10c)

If we can show that , , and , it will follow from the vanishing lemma that , which will prove the second equation in (4.5).

Comparing (4.7) and (4.10a), we see that , and then (4.10b) becomes

(4.11)

The terms of order in the asymptotic expansion of the equation yield

(4.12)

Comparing (4.12) with (4.11), it follows that . Given a matrix

let us write , where

Equation (4.7) can then be written as , and hence

(4.13)

According to (4.12), we have

(4.14)

Equations (4.13) and (4.14) imply

It only remains to prove that . The terms of order in the expansion of the equation yield

(4.15)

It follows that . On the other hand, (4.12) and (4.15) imply

We conclude that

which proves the lemma. ∎

The compatibility condition of (4.5) shows that satisfies (1.1). The proof of Theorem 1 is complete.

5. Proof of Theorem 2

Let and let be the associated solution of (1.1) defined by (2.2). Our goal is to find the asymptotics of in the sector defined by

(5.1)

where is a constant. Let

denote the right and left halves of . For conciseness, we will give the proof of the asymptotic formula (2.3) for ; the case when can be handled in a similar way but requires some (minor) changes in the arguments (see [2] for the required changes in the case of the mKdV equation).

The jump matrix defined in (2.1) involves the exponentials , where is defined by

(5.2)

Suppose . Then there are two real critical points (i.e., solutions of ) located at the points , where (see Figure 1)

As , the critical points approach at least as fast as , i.e., .

\OVP@calc
Figure 1. The critical points in the complex -plane together with the regions where (white) and (shaded).

5.1. Analytic approximation

We first decompose into an analytic part and a small remainder . Let be an integer. Let denote the contour

where

We orient to the right and let (resp. ) denote the open subset between (resp. ) and the real line, see Figure 2.

Lemma 5.1 (Analytic approximation).

There exists a decomposition

where the functions and have the following properties:

  1. For each , is defined and continuous for and analytic for .

  2. The function obeys the following estimates uniformly for :

    and

  3. The and norms of on are as uniformly for .

Proof.

See [2, Lemma 5.1]. ∎

\OVP@calc
Figure 2. The sets and and the contour .

Letting and , we obtain a decomposition of by setting

5.2. Opening of the lenses

The jump matrix enjoys the factorization

(5.3)

It follows that satisfies the RH problem in Theorem 1 if and only if the function defined by

(5.4)

satisfies the RH problem

  • is analytic in with continuous boundary values on ;

  • for ;

  • as ;

  • as ;

where the jump matrix is given by

(5.5)

Note that and obey the same symmetries (4.1) and (4.6) as and .

5.3. Local model

Let us introduce new variables and by

(5.6)

so that

(5.7)

Fix and let . Let , see Figure 3. Let denote the contour defined in (B.1) with . The map maps onto and we have , where denotes the inverse image of under this map.

\OVP@calc
Figure 3. The contour .

Let denote the th order Taylor polynomial of at , i.e.,

(5.8)

For large and fixed , the jump matrices can be approximated as follows: