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.

Riemann–Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions

2021 ◽  
pp. 2150208
Author(s):  
Bo Zhang ◽  
Engui Fan
Keyword(s):  
Boundary Conditions ◽  
Soliton Solutions ◽  
Type Equation ◽  
Spectral Parameters ◽  
Spectral Matrix ◽  
The Matrix ◽  
Nonzero Boundary ◽  
Dynamical Features ◽  
Jost Solutions ◽  
Associated Matrix

In this paper, we focus on investigating a nonlinear Schrödinger-type equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of the Schrödinger-type equation, we derive its Jost solutions with nonzero boundary conditions, and further analyze the asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. An associated matrix Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions is subsequently presented, and a formulae of [Formula: see text]-soliton solutions for the Schrödinger-type equation by solving the matrix RH problem. As an application of the [Formula: see text]-soliton formulae, we present two kinds of one-soliton solutions and three kinds of two-soliton solutions according to different distributions of spectral parameters, and dynamical features of those solutions are also further analyzed.

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.

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.

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.

The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

2021 ◽  
pp. 2150153
Author(s):  
Jian Li ◽  
Tiecheng Xia ◽  
Hanyu Wei
Keyword(s):  
Physical Meaning ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bloch Equation ◽  
Lax Pair ◽  
Integrable Equation ◽  
Motion Trajectory ◽  
The Matrix ◽  
Riemann Hilbert Problem

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

Riemann–Hilbert approach for the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation and its N-soliton solutions

2018 ◽  
Vol 32 (07) ◽  
pp. 1850088
Author(s):  
Hui Nie ◽  
Liping Lu ◽  
Xianguo Geng
Keyword(s):  
Spectral Analysis ◽  
Inverse Scattering ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
Nonlinear Schrödinger ◽  
Nonlinear Integrable Equation ◽  
Riemann Hilbert Problem

On the basis of the spectral analysis for the Lax pair, a Riemann–Hilbert problem of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is established. Using the inverse scattering transformation and the Riemann–Hilbert approach, the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is studied. As an application, N-soliton solutions of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation are obtained. In addition, some figures are given to illustrate the soliton characteristics of the nonlinear integrable equation.

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