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.

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 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.

Riemann–Hilbert Problem and Multi-Soliton Solutions of the Integrable Spin-1 Gross–Pitaevskii Equations

2019 ◽  
Vol 74 (2) ◽  
pp. 139-145 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Spectral Analysis ◽  
Nonlinear Optical ◽  
Light Transmission ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Nonlinear Phenomena ◽  
Optical Fibres ◽  
Collision Dynamics ◽  
Integrable Spin ◽  
Riemann Hilbert Problem

AbstractUnder investigation in this article is the integrable spin-1 Gross–Pitaevskii (SGP) equations, which can be used to describe light transmission in bimodal nonlinear optical fibres. The spectral analysis with 4 × 4 Lax pairs is performed for the integrable SGP equations, from which a Riemann Hilbert problem is formulated. Furthermore, N-soliton solutions of this integrable SGP equations are expressed in terms of solutions of the Riemann–Hilbert problem by using the Plemelj formulae. Finally, collision dynamics between two solitons is also analyzed. Our results can be used to enrich and explain some related nonlinear phenomena.

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.

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 Approach of The Complex Sharma-Tasso-Olver Equation and Its N-Soliton Solutions

2021 ◽  
Author(s):  
Sha Li ◽  
Tiecheng Xia ◽  
Jian Li
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
The Matrix ◽  
Relationship Of ◽  
Parameter Values ◽  
The Relationship

Abstract In this paper, we use Riemann-Hilbert method to study the N-soliton solutions of the complex Sharma-Tasso-Olver(cSTO) equation. And then, based on analyzing the spectral problem of the Lax pair, the matrix Riemann-Hilbert problem for this integrable equation can be constructed, the N-soliton solutions about this system are given explicitly under the relationship of scattering matrix. At last, under the condition that some specifific parameter values are given, the three-dimensional diagram of the 2-soliton solution and the trajectory of the soliton solution will be simulated.

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