Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions

2020 ◽  
Vol 402 ◽  
pp. 132170 ◽  
Author(s):  
Guoqiang Zhang ◽  
Zhenya Yan
Keyword(s):  
Boundary Conditions ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Scattering Transforms

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.

Inverse scattering method for the Kundu-Eckhaus equation with zero/nonzero boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Guixian Wang ◽  
Xiu-Bin Wang ◽  
Bo Han ◽  
Qi Xue
Keyword(s):  
Boundary Conditions ◽  
Inverse Scattering ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Higher Order ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Nonzero Boundary ◽  
Wave Phenomena

Abstract In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena.

scholarly journals Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 512
Author(s):  
Liming Ling ◽  
Wen-Xiu Ma
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Identity Matrix ◽  
Nonlocal Symmetry ◽  
Scattering Problems ◽  
Spectral Problems ◽  
Symmetry Reductions ◽  
Jump Matrix ◽  
De Vries ◽  
Scattering Transforms

This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms.

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