Riemann-Hilbert problem for the sextic nonlinear Schr\"{o}dinger equation with non-zero boundary conditions

TL;DR

This paper formulates and solves a Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with non-zero boundary conditions, analyzing spectral properties and soliton solutions using complex analysis and inverse scattering techniques.

Contribution

It introduces a new Riemann-Hilbert framework for the sextic NLS with non-zero boundary conditions, including spectral analysis and explicit soliton solutions.

Findings

Derived explicit soliton solutions for different pole cases.

Analyzed localized structures and dynamic behaviors of solitons.

Established a generalized Riemann-Hilbert problem for the equation.

Abstract

We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr\"{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems. Based on a new complex plane, the direct scattering problem perform a detailed analysis of the analytical, asymptotic and symmetry properties of the Jost functions and the scattering matrix. Then, a generalized Riemann-Hilbert problem (RHP) is successfully established from the results of the direct scattering transform. In the inverse scattering problem, we discuss the discrete spectrum, residue condition, trace formula and theta condition under simple poles and double poles respectively, and further solve the solution of a generalized RHP. Finally, we derive the solution of the equation for the cases of different poles without reflection potential. In addition, we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.