Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schr\"{o}dinger equation with non-vanishing boundary conditions

TL;DR

This paper develops a Riemann-Hilbert framework to analyze the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions, deriving soliton and breather solutions and exploring their behaviors.

Contribution

It introduces a novel Riemann-Hilbert approach for the equation with non-vanishing boundaries, including double poles, and systematically solves the inverse scattering problem.

Findings

Explicit soliton and breather solutions are derived.

The impact of parameters on solution behaviors is analyzed graphically.

The method handles simple and double poles in the scattering data.

Abstract

In this work, we consider the generalized variable-coefficient nonlinear Schr\"{o}dinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an appropriate Riemann surface and uniformization coordinate variable, we first convert the double-valued functions which occur in the process of direct scattering to single-value functions. Then, we establish the direct scattering problem via analyzing the analyticity, symmetries and asymptotic behaviors of Jost functions and scattering matrix derived from Lax pairs of the equation. Based on these results, a generalized Riemann-Hilbert problem is successfully established for the equation. The discrete spectrum and residual conditions, trace foumulae and theta conditions are investigated systematically including the simple poles case and double poles case. Moreover, the inverse scattering problem is solved via the Riemann-Hilbert approach. Finally, under the condition of reflection-less potentials, the soliton and breather solutions are well derived. Via evaluating the impact of each parameters, some interesting phenomena of these solutions are analyzed graphically.