Inverse scattering transforms and soliton solutions of nonlocal reverse‐space nonlinear Schrödinger hierarchies

2020 ◽  
Vol 145 (3) ◽  
pp. 563-585
Author(s):  
Wen‐Xiu Ma ◽  
Yehui Huang ◽  
Fudong Wang
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Nonlinear Schrödinger ◽  
Scattering Transforms

Riemann–Hilbert approach and multi-soliton solutions of a variable-coefficient fifth-order nonlinear Schrödinger equation with N distinct arbitrary-order poles

2021 ◽  
pp. 2150194
Author(s):  
Zhi-Qiang Li ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang ◽  
Jin-Jie Yang
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Arbitrary Order ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Fifth Order

Based on inverse scattering transformation, a variable-coefficient fifth-order nonlinear Schrödinger equation is studied through the Riemann–Hilbert (RH) approach with zero boundary conditions at infinity, and its multi-soliton solutions with [Formula: see text] distinct arbitrary-order poles are successfully derived. By deriving the eigenfunction and scattering matrix, and revealing their properties, a RH problem (RHP) is constructed based on inverse scattering transformation. Via solving the RHP, the formulae of multi-soliton solutions are displayed when the reflection coefficient possesses [Formula: see text] distinct arbitrary-order poles. Finally, some appropriate parameters are selected to analyze the interaction of multi-soliton solutions graphically.

Inverse scattering transforms for non-local reverse-space matrix non-linear Schrödinger equations

2021 ◽  
pp. 1-21
Author(s):  
WEN-XIU MA ◽  
YEHUI HUANG ◽  
FUDONG WANG
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Problems ◽  
Plemelj Formula ◽  
Non Linear ◽  
Non Local ◽  
Space Matrix ◽  
Scattering Transforms

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

scholarly journals Dynamics of soliton solutions of the fifth-order nonlinear Schrödinger equation via the Riemann-Hilbert approach

2021 ◽  
Author(s):  
Jin-Jie Yang ◽  
Shou-Fu Tian ◽  
TIan-Tian Zhang ◽  
Xiao-Li Wang
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Scattering Data ◽  
Scattering Process ◽  
Nonlinear Schrödinger ◽  
Propagation Behavior ◽  
Direct Scattering ◽  
The One ◽  
New Phenomena ◽  
Fifth Order

The theory of inverse scattering is developed to investigate the initial-value problem for the fifth-order nonlinear Schrödinger (foNLS) equation under the zero boundary conditions at infinity. The spectral analysis is performed in the direct scattering process, including the establishment of the analytical, asymptotic and symmetric properties of the scattering matrix and the Jost functions. In the inverse scattering process, a suitable Riemann-Hilbert (RH) problem is successfully established by using the modified eigenfunctions and scattering data, and the relationship between the potential function and the solution of the RH problem is successfully established. In order to further analyze the propagation behavior of the solutions of the foNLS equation, we present some new phenomena of studying the one-, two-, and three- soliton solutions corresponding to simple zeros in scattered data. Finally, we also analyze the one- and two-soliton solutions corresponding to double zeros.

scholarly journals Prolongation Structures and N-Soliton Solutions for a New Nonlinear Schrödinger-Type Equation via Riemann-Hilbert Approach

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yuxin Lin ◽  
Yong Fang ◽  
Huanhe Dong
Keyword(s):  
Spectral Problem ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Type Equation ◽  
Lax Pair ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
Matrix Spectral Problem

In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a 2×2 matrix spectral problem, is derived. Depending on the analysis of red the spectral problem, a R-H problem of the NLST equation is formulated. Furthermore, through a specific R-H problem with the vanishing scattering coefficient, N-soliton solutions of the NLST equation are expressed explicitly. Moreover, a few key differences are presented, which exist in the implementation of the inverse scattering transform for NLST equation and cubic nonlinear Schrödinger (NLS) equation. Finally, the dynamic behaviors of soliton solutions are shown by selecting appropriate spectral parameter λ, respectively.

Inverse scattering for nonlocal reverse-space multicomponent nonlinear Schrödinger equations

2021 ◽  
Vol 35 (04) ◽  
pp. 2150051
Author(s):  
Wen-Xiu Ma ◽  
Yehui Huang ◽  
Fudong Wang
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Problems ◽  
Nonlinear Schrödinger ◽  
Plemelj Formula ◽  
Nonlinear Schrodinger Equations ◽  
Generalized Matrix

The paper aims to discuss nonlocal reverse-space multicomponent nonlinear Schrödinger equations and their inverse scattering transforms. The inverse scattering problems are analyzed by means of Riemann–Hilbert problems, and Gelfand–Levitan–Marchenko-type integral equations for generalized matrix Jost solutions are determined by the Sokhotski–Plemelj formula. Soliton solutions are generated from the reflectionless transforms associated with zeros of the Riemann–Hilbert problems.

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