# Inverse Scattering Transformation for the Fokas–Lenells Equation with Nonzero Boundary Conditions

^{*}

^{*}Corresponding author. Email: faneg@fudan.edu.cn

- DOI
- 10.2991/jnmp.k.200922.003How to use a DOI?
- Keywords
- Fokas–Lenells equation; nonzero boundary conditions; inverse scattering transformation; Riemann–Hilbert problem; N-soliton solution
- Abstract
In this article, we focus on the inverse scattering transformation for the Fokas–Lenells (FL) equation with nonzero boundary conditions via the Riemann–Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity, symmetry and asymptotic behavior of the Jost solutions and scattering matrix are discussed in detail. With these results, we further present a generalized RH problem, from which a reconstruction formula between the solution of the FL equation and the Riemann–Hilbert problem is obtained. The

*N*-soliton solutions of the FL equation is obtained by solving the RH problem.- Copyright
- © 2020 The Authors. Published by Atlantis Press B.V.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. INTRODUCTION

The Nonlinear Schrödinger (NLS) equation

*ν*= ±1 is a significant mathematical and physical model for the optical fibers, deep water waves and plasma physics [3]. The NLS equation is a completely integrable system which admits Lax pair and bi-Hamiltonian formulation [21,22].

In the late 70s, standard bi-Hamiltonian formulation was used to obtain an integrable generalization of a given equation [7]. For instance, the Camassa–Holm equation is derived from the KdV equation via the bi-Hamiltonian structure [6], and the same mathematical trick can be applied to the NLS equation yields Fokas–Lenells (FL) equation [8]

If let *α* = *γ*/*ν* > 0 and

*α*,

*β*> 0. In this paper, we consider the focusing case with

*σ*= −1.

In recent years, much work has been done on the FL equation (1.3). For example, Lax pair for FL equation was obtained via the bi-Hamiltonian structure by Fokas and Lenells [8]. They further considered the initial-boundary problem for the FL equation (1.3) on the half-line by using the Fokas unified method [9]. The dressing method is applied to obtain an explicit formula for bright-soliton and dark soliton solutions for Eq. (1.3) [11,15]. The bilinear method was used to obtain bright and dark soliton solution are obtained [12,13]. The Darboux transformation is used to obtain rogue waves of the FL equation [10,19]. The Deift–Zhou nonlinear steepest decedent method was used to analyze long-time asymptotic behavior for FL equation with decaying initial value [18]. Riemann–Hilbert (RH) approach was adopted to construct explicit soliton solutions under zero boundary conditions [1]. As far as we know, the soliton solutions for the FL equation (1.3) with Nonzero Boundary Conditions (NZBCs) have not been reported. In this paper, we apply RH approach to study the inverse transformation of the FL equation (1.3) with the following NZBCs

*u*(

*x*,

*t*) −

*q*

_{±}∈

*L*

^{1}(

*R*±); with respect to

*x*for all

*t*≥ 0. In next section, we will see that this kind of NZBCs (1.4) avoids the discussion on a multi-valued function as the case of the NLS equation [3].

The inverse scattering transform is an important method to study important nonlinear wave equations with Lax pair such as the NLS equation, the modified KdV equation, Sine-Gordan equation [5,14]. As an improved version of inverse scattering transform, the RH method has been widely adopted to solve nonlinear integrable systems [2,4,16,17,20,23].

The paper is organized as follows. In Section 2, by introducing appropriate transformations, we change the asymptotic boundary conditions (1.4) into constant boundary conditions. Furthermore, we analyze the analyticity, symmetry and asymptotic behavior of eigenfunctions and scattering matrix associated with the Lax pair. In Section 3, a generalized RH problem for the FL equation is constructed, and the distribution of discrete spectrum and residue conditions associated with RH problem are discussed. Based on these results, we reconstruct the potential function from the solution of the RH problem. In Section 4, we give the *N*-soliton solutions via solving RH problem under reflectionless case.

## 2. THE DIRECT SCATTERING WITH NZBCS

## 2.1. Jost Solutions

It is well-known that the FL equation (1.3) admits a Lax pair

In order to invert the nonzero boundary conditions into the constant boundary conditions, we introduce a transformation for eigenfunctions and potentials.

### Theorem 2.1.

By transformation

And Eq. (2.3) is the compatibility condition of the Lax pair

*Proof.* Suppose the transformation is

Substituting it into (1.3) yields

And the Lax pair becomes

Take *A* = 2*C* and *B* = 2*D* in the above equation and consider the limit as *x* → ±∞, one gets

These two matrices are proportional if

Moreover, *T*_{±} and *X*_{±} are proportional

To diagonalize the matrices *X*_{±} and *T*_{±} and further obtain the Jost solutions, we need to get the eigenvalues and eigenvector matrices of them. Direct calculation shows that the eigenvalues of the matrices *X*_{±} are
*T*_{±} are ±*iηλ*, where

With the above results, the matrices can be diagonalized as:

Thus, the asymptotic Lax pair

Therefore, the asymptotic of the eigenfunctions *φ*_{±} are

Define the Jost solutions as

## 2.2. Scattering Matrix

The functions *φ*_{±} are both fundamental matrix solutions of the Lax pair (2.5), thus there exists a matrix *S*(*k*) that only depends on *k*, such that

*S*(

*k*) is called scattering matrix. Columnwise, Eq. (2.15) reads

Since tr(*X*) = tr(*T*) = 0, according to the Abel’s formula, we have

*φ*

_{±}in the following way

## 2.3. Asymptotic Analysis

To properly construct the Riemann–Hilbert problem, we need to consider the asymptotic behavior of eigenfunctions and scattering matrix as *k* → ∞ and *k* → 0.

## 2.3.1. Asymptotic as k → ∞

Consider a solution of (2.13) of the form

*k*, we get the comparison results:

*x*-part:

*t*-part:

Based on these results, we derive that

Note that the FL equation (2.3) admits the conservation law

Therefore, we obtain the limit of the Jost solutions as *k* → ∞:

Define

In addition, the asymptotic property of *φ*_{±} follows:

Consider the wronskians expression of the scattering coefficients (2.17), we find that

## 2.3.2. Asymptotic as k → 0

We first assume that *J*_{±} admit a Lorrent expansion

Substituting into (2.13) and comparing the coefficients of the same order of *k* gives that

*x*-part:

*t*-part:

It is easy to check that

*x*. This implies that the following limit

*x*∈

*R*. While the following limit

*k*∈

*C*. Thus the following limit is commutative, and using (2.32) and (2.33) yields

The asymptotics of functions *φ*_{±} and *μ*_{±} can be obtained by transformations (2.12) and (2.26). Consider the wronskian expressions of scattering coefficients and note the boundary conditions, we find that

## 2.4. Analyticity

Noticing that

Under the transformation (2.26), we write the Lax pair as

We rewrite the Eq. (2.35) into a full differential form

*μ*

_{±}can be formally integrated to obtain the integral equations for the eigenfunctions:

where

By using a similar way to Appendix A in Biondini and Kovačič [1], under mild integrability conditions on the potential, the eigenfunctions (2.37) and (2.38) can be analytically extended in the complex *k*-plane into the following regions:

*μ*

_{±}= (

*μ*

_{±}

_{,1,}

*μ*

_{±}

_{,2}), the subscript 1, 2 denote the first and second column of

*μ*

_{±}.

Apparently, the functions *φ*_{±} and *μ*_{±} share the same analyticity, hence from the wronskians expression of the scattering coefficients, we know that *s*_{11} is analytic in *D*^{−}, and *s*_{22} is analytic in *D*^{+}.

## 2.5. Symmetry

To investigate the discrete spectrum and residue conditions in the Riemann–Hilbert problem, one needs to analyze the symmetric property for the solutions *φ*_{±} and the scattering matrix *S*(*k*).

### Theorem 2.2.

The Jost eigenfunctions satisfy the following symmetric relations

*Proof.* We only prove (2.40) and (2.42), (2.41) and (2.43) can be shown in a similar way. The functions *φ*_{±} are the solutions of the spectral problem

Conjugating and multiplying left by *σ* on both sides of the equation gives

Note that *σX*^{*}(*k*^{*}) *= X*(*k*)*σ*, hence

For the scattering matrix, conjugating on the both sides of Eq. (2.15) leads to

Elementwise, Eqs. (2.40)–(2.43) read as

*μ*

_{±}, we derive

## 3. THE INVERSE SCATTERING WITH NZBCS

## 3.1. Generalized Riemann–Hilbert Problem

We define the two matrices

*D*

^{+},

*D*

^{−}respectively, and admit asymptotic

By using (2.49) and (3.1), we rewrite (2.16) and get a generalized Riemann–Hilbert problem

where the jump matrix

## 3.2. Discrete Spectrum and Residue Conditions

Suppose that *s*_{11}(*k*) has *N* simple zeroes in *D*^{+} ∩{Im *k* < 0} denoted by *k _{n}*,

*n*= 1, 2, ...,

*N*. Owing to the symmetries in (2.46), there exists

*k*-plane as shown in Figure 2.

Next we derive the residue conditions will be needed for the RH problem. If *s*_{11}(*k*) = 0 at *k* = *k _{n}* the eigenfunctions

*φ*

_{+,1}and

*φ*

_{−,2}at

*k*=

*k*must be proportional

_{n}*b*is an arbitrary constant independent on

_{n}*x*,

*t*. Under the transformation (2.49), there exists linear relation for

*μ*

_{±}

Thus we get

*k*= −

*k*, substituting (2.48) into (3.8) leads to

_{n}Similarly, the residue conditions at

where

Recall the definition of *M*^{±}, there follows

## 3.3. Reconstruction Formula for the Potential

To solve the Riemann–Hilbert problem (3.3), one needs to regularize it by subtracting out the asymptotic behaviour and the pole contribution. Hence, we rewrite Eq. (3.3) as

*M*is

Comparing the (1, 2)-element on the both sides of Eq. (2.20) yields

Recall the transformation (2.2), we know

Substituting (3.19) and (3.16) into (3.21), we obtain the reconstruction formula for potential

## 4. REFLECTIONLESS POTENTIALS

Now we consider the potential *u*(*x*, *t*) for which the reflection coefficient *ρ*(*k*) vanishes identically, that is, *G* = 0. In this case, Eq. (3.22) reads as

To obtain the expression of the term

Define

Then (4.3) reduces to

substituting (4.5) into (4.4) yields

Introducing notations

By standard Crammer rule, the system (4.7) is the solution of

Note that

Therefore, we obtain a compact solution:

*N*+ 1) × (

*N*+ 1) matrix

*H*

^{aug}is

## 5. TRACE FORMULA AND THETA CONDITION

Define

*D*

^{−}and

*D*

^{+}, respectively. Moreover,

*β*

^{+}

*β*

^{−}=

*s*

_{11}(

*k*)

*s*

_{22}(

*k*). Note that d

*et*

*S*(

*k*) =

*s*

_{11}

*s*

_{22}−

*s*

_{21}

*s*

_{12}= 1, this implies

Taking logarithms leads to

Substituting into (5.1), we obtain the trace formula

Under reflectionless condition, they reduce to

Taking limit as *k* → 0 for (5.4) leads to

*β*is a positive constant, then we obtain the theta condition

## 6. ONE-SOLITON SOLUTION

As an application of the formula (4.10) of *N*-soliton solution, we construct one-soliton solution for the FL equation, which corresponds to *N* = 1. Then Eq. (4.10) becomes

*k*

_{1}is an eigenvalue,

*C*

_{1}is an arbitrary constant and

## CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

## ACKNOWLEDGMENT

This work is supported by the

## REFERENCES

### Cite this article

TY - JOUR AU - Yi Zhao AU - Engui Fan PY - 2020 DA - 2020/12/10 TI - Inverse Scattering Transformation for the Fokas–Lenells Equation with Nonzero Boundary Conditions JO - Journal of Nonlinear Mathematical Physics SP - 38 EP - 52 VL - 28 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.k.200922.003 DO - 10.2991/jnmp.k.200922.003 ID - Zhao2020 ER -