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

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.