INVERSE SCATTERING TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

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Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation

Communications on Pure and Applied Analysis ◽

10.3934/cpaa.2021178 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Yuan Li ◽

Shou-Fu Tian

Keyword(s):

Inverse Scattering ◽

Hilbert Problem ◽

Scattering Problem ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Scattering Data ◽

Symmetric Relation ◽

Hirota Equation ◽

Direct Scattering ◽

Scattering Transform

<p style='text-indent:20px;'>In this work, we study the inverse scattering transform of a nonlocal Hirota equation in detail, and obtain the corresponding soliton solutions formula. Starting from the Lax pair of this equation, we obtain the corresponding infinite number of conservation laws and some properties of scattering data. By analyzing the direct scattering problem, we get a critical symmetric relation which is different from the local equations. A novel left-right Riemann-Hilbert problem is proposed to develop the inverse scattering theory. The potentials are recovered and the pure soliton solutions formula is obtained when the reflection coefficients are zero. Based on the zero types of scattering data, nine types of soliton solutions are obtained and three typical types are described in detail. In addition, some dynamic behaviors are given to illustrate the soliton characteristics of the space symmetric nonlocal Hirota equation.</p>

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The elliptic sinh-Gordon equation in a semi-strip

Advances in Nonlinear Analysis ◽

10.1515/anona-2016-0206 ◽

2017 ◽

Vol 8 (1) ◽

pp. 533-544 ◽

Cited By ~ 2

Author(s):

Guenbo Hwang

Keyword(s):

Gordon Equation ◽

Hilbert Problem ◽

Lax Pair ◽

Inverse Scattering Transform ◽

Boundary Values ◽

Spectral Functions ◽

Scattering Transform ◽

The Matrix ◽

Fokas Method ◽

Riemann Hilbert Problem

Abstract We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

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