N-soliton solution for a higher-order Chen–Lee–Liu equation with nonzero boundary conditions

2019 ◽  
Vol 34 (04) ◽  
pp. 2050054 ◽  
Author(s):  
Yi Zhao ◽  
Engui Fan
Keyword(s):  
Boundary Conditions ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Higher Order ◽  
Third Order ◽  
Third Order Dispersion ◽  
Nonzero Boundary ◽  
Riemann Hilbert Problem

In this paper, the Riemann–Hilbert approach is applied to investigate a higher-order Chen–Lee–Liu equation with third-order dispersion and quintic nonlinearity terms. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a generalized Riemann–Hilbert problem associated with Chen–Lee–Liu equation with nonzero boundary conditions is constructed. Further, the [Formula: see text]-soliton solution is found by solving the generalized Riemann–Hilbert problem. As an illustrative example, two kinds of one-soliton solutions with different forms of parameters are obtained.

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 Multiple-Pole Solutions to a Semidiscrete Modified Korteweg-de Vries Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Zhixing Xiao ◽  
Kang Li ◽  
Junyi Zhu
Keyword(s):  
Soliton Solution ◽  
Vries Equation ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Higher Order ◽  
Symmetry Condition ◽  
De Vries ◽  
Riemann Hilbert Problem

Multiple-pole soliton solutions to a semidiscrete modified Korteweg-de Vries equation are derived by virtue of the Riemann-Hilbert problem with higher-order zeros. A different symmetry condition is introduced to build the nonregular Riemann-Hilbert problem. The simplest multiple-pole soliton solution is presented. The dynamics of the solitons are studied.

Riemann–Hilbert approach and N-soliton solutions for the Chen–Lee–Liu equation

2019 ◽  
Vol 33 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ming-Jun Xu ◽  
Tie-Cheng Xia ◽  
Bei-Bei Hu
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Potential Matrix ◽  
Riemann Hilbert Problem

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions.

On a Riemann–Hilbert problem for the NLS-MB equations

2021 ◽  
pp. 2150420
Author(s):  
Leilei Liu ◽  
Weiguo Zhang ◽  
Jian Xu
Keyword(s):  
Hilbert Problem ◽  
Soliton Solutions ◽  
Coupled System ◽  
Soliton Interaction ◽  
Nls Equation ◽  
Nonzero Boundary ◽  
Complex Symmetric ◽  
Complex Phenomena ◽  
Symmetric Relations ◽  
Riemann Hilbert Problem

In this paper, we study a coupled system of the nonlinear Schrödinger (NLS) equation and the Maxwell–Bloch (MB) equation with nonzero boundary conditions by Riemann–Hilbert (RH) method. We obtain the formulae of the simple-pole and the multi-pole solutions via a matrix Riemann–Hilbert problem (RHP). The explicit form of the soliton solutions for the NLS-MB equations is obtained. The soliton interaction is also given. Furthermore, we show that the multi-pole solutions can be viewed as some proper limits of the soliton solutions with simple poles, and the multi-pole solutions constitute a novel analytical viewpoint in nonlinear complex phenomena. The advantage of this way is that it avoids solving the complex symmetric relations and repeatedly solving residue conditions.

scholarly journals The Riemann–Hilbert approach to focusing Kundu–Eckhaus equation with non-zero boundary conditions

2020 ◽  
Vol 34 (30) ◽  
pp. 2050332
Author(s):  
Li-Li Wen ◽  
En-Gui Fan
Keyword(s):  
Riemann Surface ◽  
Boundary Conditions ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Spectral Matrix ◽  
First Order ◽  
Dynamical Features ◽  
Jost Solutions ◽  
Riemann Hilbert Problem

In this paper, we investigate the focusing Kundu–Eckhaus equation with non-zero boundary conditions. An appropriate two-sheeted Riemann surface is introduced to map the spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of Kundu–Eckhaus equation, two kinds of Jost solutions are constructed. Further, their asymptotic, analyticity, symmetries as well as spectral matrix are analyzed in detail. It is shown that the solution of the Kundu–Eckhaus equation with non-zero boundary conditions can be characterized with a matrix Riemann–Hilbert problem. Then a formula of [Formula: see text]-soliton solutions is derived by solving the Riemann–Hilbert problem. As applications of the [Formula: see text]-soliton formula, the first-order explicit soliton solutions with different dynamical features are obtained and analyzed.

scholarly journals N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 826 ◽  
Author(s):  
Yuxin Lin ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Perturbation Theory ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nls Equation ◽  
Jump Matrix ◽  
Potential Matrix ◽  
Relationship Of ◽  
Riemann Hilbert Problem

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P − ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived.

scholarly journals Riemann-Hilbert Problem with Higher Order Poles for Self-Induced Transparency with Degenerate Energy Levels

1988 ◽  
Vol 41 (6) ◽  
pp. 735 ◽  
Author(s):  
A Roy Chowdhury ◽  
Mrityunjoy De
Keyword(s):  
Inverse Problem ◽  
Hilbert Problem ◽  
Energy Levels ◽  
Soliton Solutions ◽  
Higher Order ◽  
Second Order ◽  
Induced Transparency ◽  
Riemann Hilbert Problem

We have analysed the inverse problem for self-induced transparency with degenerate energy levels (DSIl) via the Riemann-Hilbert methodology with second order poles. The required formulae are deduced in detail and the corresponding soliton solutions are obtained. It is noticed that the profile of the single soliton is not of the usual sech type.

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