Riemann-Hilbert problem for the nonlinear Schr\"{o}dinger equation with multiple high-order poles under nonzero boundary conditions

TL;DR

This paper develops a Riemann-Hilbert approach to analyze the focusing nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions, deriving explicit soliton solutions and their propagation behaviors.

Contribution

It introduces a novel Riemann-Hilbert framework for high-order poles in the NLS equation, providing explicit solutions and analyzing their dynamics.

Findings

Explicit soliton solutions for high-order poles are obtained.

Propagation behaviors of third-order pole solitons are demonstrated.

A systematic method for handling multiple high-order poles is developed.

Abstract

The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr\"{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues at the simple- and the second-poles. Further, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole is given as example.