The general fifth-order nonlinear Schr\"odinger equation with nonzero boundary conditions: Inverse scattering transform and multisoliton solutions

TL;DR

This paper develops the inverse scattering transform for the fifth-order nonlinear Schrödinger equation with nonzero boundary conditions, deriving explicit multisoliton solutions and analyzing their dynamics.

Contribution

It introduces a systematic Riemann-Hilbert problem approach for the equation with NZBCs and constructs explicit multisoliton solutions, including simple and double poles.

Findings

Explicit multisoliton solutions with NZBCs are obtained.

The inverse scattering transform is formulated via a matrix Riemann-Hilbert problem.

The dynamics of solutions are graphically analyzed.

Abstract

Under investigation in this work is the inverse scattering transform of the general fifth-order nonlinear Schr\"{o}dinger equation with nonzero boundary conditions (NZBCs), which can be reduced to several integrable equations. Firstly, a matrix Riemann-Hilbert problem for the equation with NZBCs at infinity is systematically investigated.Then the inverse problems are solved through the investigation of the matrix Riemann-Hilbert problem. Thus, the general solutions for the potentials, and explicit expressions for the reflection-less potentials are well constructed. Furthermore, the trace formulae and theta conditions are also presented. In particular, we analyze the simple-pole and double-pole solutions for the equation with NZBCs. Finally, the dynamics of the obtained solutions are graphically discussed. These results provided in this work can be useful to enrich and explain the related nonlinear wave phenomena in nonlinear fields.