Inverse scattering transform of an extended nonlinear Schr\"{o}dinger equation with nonzero boundary conditions and its multisoliton solutions

TL;DR

This paper develops an inverse scattering transform method for an extended nonlinear Schrödinger equation with nonzero boundary conditions, deriving multisoliton solutions and analyzing their properties in nonlinear wave contexts.

Contribution

It introduces a systematic Riemann-Hilbert problem approach for the equation with nonzero boundaries and constructs explicit multisoliton solutions, including simple and double-pole cases.

Findings

Explicit multisoliton solutions derived

Reflection-less potentials explicitly expressed

Graphical analysis of soliton characteristics

Abstract

Under investigation in this work is an extended nonlinear Schr\"{o}dinger equation with nonzero boundary conditions, which can model the propagation of waves in dispersive media. Firstly, a matrix Riemann-Hilbert problem for the equation with nonzero boundary conditions at infinity is systematically discussed. Then the inverse problems are solved through the investigation of the matrix Riemann-Hilbert problem. Therefore, the general solutions for the potentials, and explicit expressions for the reflection-less potentials are presented. Furthermore, we construct the simple-pole and double-pole solutions for the equation. Finally, the remarkable characteristics of these solutions are graphically discussed. Our results should be useful to enrich and explain the related nonlinear wave phenomena in nonlinear fields.