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.

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.

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.

The multi-triple-pole solitons for the focusing mKdV hierarchy with nonzero boundary conditions

2021 ◽  
pp. 2150483
Author(s):  
Weifang Weng ◽  
Zhenya Yan
Keyword(s):  
Boundary Conditions ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Scattering Coefficients ◽  
Dark Solitons ◽  
Scattering Transform ◽  
Nonzero Boundary ◽  
Trace Formulae ◽  
Fifth Order ◽  
Mkdv Hierarchy

In this paper, the general triple-pole multi-soliton solutions are proposed for the focusing modified Korteweg–de Vries (mKdV) equation with both nonzero boundary conditions (NZBCs) and triple zeros of analytical scattering coefficients by means of the inverse scattering transform. Furthermore, we also give the corresponding trace formulae and theta conditions. Particularly, we analyze some representative reflectionless potentials containing the triple-pole multi-dark-anti-dark solitons and breathers. The idea can also be extended to the whole mKdV hierarchy (e.g. the fifth-order mKdV equation, and third-fifth-order mKdV equation) with NZBCs and triple zeros of analytical scattering coefficients. Moreover, these obtained triple-pole solutions can also be degenerated to the triple-pole soliton solutions with zero boundary conditions.

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 Multi-Soliton Solutions of the N-Component Nonlinear Schrodinger Equations via Riemann-Hilbert Approach*

2021 ◽  
Author(s):  
Yan Li ◽  
Jian Li ◽  
Ruiqi Wang
Keyword(s):  
Soliton Solutions ◽  
Lax Pair ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Spectral Parameters ◽  
Nonlinear Schrödinger ◽  
Jump Matrix ◽  
Nonlinear Schrodinger Equations ◽  
The Matrix

Abstract In this paper, we utilize the Riemann-Hilbert approach to discuss multi-soliton solutions of the N-component nonlinear Schrodinger equations. Firstly, by transformed Lax pair, we construct the matrix valued functions P1,2 that satisfy the analyticity and normalization and the corresponding jump matrix can be determined. Then, in the reflectionless case, we get the multi-soliton solutions ql(l = 1, ..., N) of the N-component nonlinear Schrodinger equations, which are related to spectral parameters. Particularly, the 2-soliton solutions q1, q2 and q3 of the three-component nonlinear Schrodinger equations are given and the corresponding 2-soliton diagrams are drawn.

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