Mixed single, double, and triple poles solutions for the space-time shifted nonlocal DNLS equation with nonzero boundary conditions via Riemann--Hilbert approach

TL;DR

This paper develops a Riemann--Hilbert approach to derive and analyze mixed pole soliton solutions for a nonlocal DNLS equation with nonzero boundary conditions, revealing complex soliton interactions and effects of shift parameters.

Contribution

It introduces the first Riemann--Hilbert framework for the space-time shifted nonlocal DNLS equation, deriving explicit mixed pole solutions and analyzing their dynamic behaviors.

Findings

Derived explicit mixed pole soliton solutions with reflectionless potentials.

Graphically demonstrated complex interactions of multi-soliton, breather, and soliton-breather solutions.

Analyzed the influence of shift parameters on soliton dynamics through simulations.

Abstract

In this paper, we investigate the space-time shifted nonlocal derivative nonlinear Schr\"{o}dinger (DNLS) equation under nonzero boundary conditions using the Riemann--Hilbert (RH) approach for the first time. To begin with, in the direct scattering problem, we analyze the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions. Subsequently, we examine the coexistence of $N$-single, $N$-double, and $N$-triple poles in the inverse scattering problem. The corresponding residue conditions, trace formulae, $\theta$ condition, and symmetry relations of the norming constants are obtained. Moreover, we derive the exact expression for the mixed single, double, and triple poles solutions with the reflectionless potentials by solving the relevant RH problem associated with the space-time shifted nonlocal DNLS equation. Furthermore, to further explore the remarkable characteristics of soliton solutions, we graphically illustrate the dynamic behaviors of several representative solutions, such as three-soliton, two-breather, and soliton-breather solutions. Finally, we analyze the effects of shift parameters through graphical simulations.