The Riemann-Hilbert approach for the nonlocal derivative nonlinear Schr\"odinger equation with nonzero boundary conditions

TL;DR

This paper applies the Riemann-Hilbert method to analyze the nonlocal derivative nonlinear Schrödinger equation with nonzero boundary conditions, deriving explicit solutions and exploring their dynamics.

Contribution

It develops a comprehensive Riemann-Hilbert framework for the nonlocal derivative NLS equation, including solution construction and dynamic analysis.

Findings

Constructed explicit soliton and breather solutions.

Derived trace and reconstruction formulas.

Analyzed solution dynamics through simulations.

Abstract

In this paper, the nonlocal reverse space-time derivative nonlinear Schr\"odinger equation under nonzero boundary conditions is investigated using the Riemann-Hilbert (RH) approach. The direct scattering problem focuses on the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions, leading to the construction of the corresponding RH problem. Then, in the inverse scattering problem, the Plemelj formula is employed to solve the RH problem. So the reconstruction formula, trace formulae, $\theta$ condition, and exact expression of the single-pole and double-pole solutions are obtained. Furthermore, dark-dark solitons, bright-dark solitons, and breather solutions of the reverse space-time derivative nonlinear Schr\"odinger equation are presented along with their dynamic behaviors summarized through graphical simulation.