Riemann-Hilbert approach and N-soliton solutions for a new four-component nonlinear Schrödinger equation

Xin Mei Zhou,Shou Fu Tian,Jin Jie Yang,Jin Jin Mao

doi:10.22541/au.158379368.81935377

Abstract

A new four-component nonlinear Schrodinger equation is first proposed in this work and studied by Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a $5\times5$ matrix spectral problem for the four-component nonlinear Schrodinger equation. Then based on the Lax pair, we analyze the spectral problem and the analytical properties of the Jost functions, from which the Riemann-Hilbert problem of the equation is successfully established. Moreover, we obtain the $N$-soliton solutions of the equation by solving the Riemann-Hilbert problem without reflection. Finally, we derive two special cases of the solutions to the equation for $N=1$ and $N=2$, and the local structure and dynamic behavior of the one-and two-soliton solutions are analyzed graphically.

Highlights

The nonlinear Schrodinger equation (NLS) is an important integrable model
Preprint submitted to Journal of LATEX Templates ways to find solutions for nonlinear integrable models, including inverse scattering transform [1], Darboux transform [2], Hirota bilinear method [3], Lie group method [4], etc
For second-order spectral problems, inverse scattering theory is equivalent to Riemann-Hilbert (RH) approach, but for higher-order spectral problems the development of inverse scattering theory is not perfect, part of the inverse scattering problem needs to be transformed into RH problem

Summary

Introduction

The nonlinear Schrodinger equation (NLS) is an important integrable model. It is closely related to many nonlinear problems in theoretical physics such as nonlinear optics and ion acoustic waves of plasmas. Inverse scattering transform method is one of the most effective tools for solving the initial value problem of nonlinear integrable systems to get the soliton solutions. We first propose an interesting equation named by a new four-component nonlinear Schrodinger (FCNLS) equations iq1t + q1xx − 2[a11|q1|2 + a22|q2|2 + a33|q3|2 + a44|q4|2. The FCNLS equation includes group velocity dispersion, self-phase modulation, cross-phase modulation and paired tunnel modulation This equation can be reduced to the threecomponent nonlinear Schrodinger equation (1.3) given by iq1t q1xx. The main purpose of this work is to study the RH problem for the FCNLS equation (1.2) by first deriving its Lax pair, and obtain it’s N-soliton solutions. We derive a Lax pair associated with a 5×5 matrix spectral problem for the FCNLS equation (1.2).

The Lax Pair and eigenfunction

Riemann-Hilbert problem

Multi-soliton solutions

Conclusions and discussions