Inverse scattering transform for the discrete nonlocal PT symmetric nonlinear Schr\"odinger equation with nonzero boundary conditions

TL;DR

This paper develops an inverse scattering transform method for a discrete nonlocal PT symmetric nonlinear Schrödinger equation with nonzero boundary conditions, analyzing different cases and deriving soliton solutions.

Contribution

It introduces a comprehensive inverse scattering framework for the discrete nonlocal PT symmetric NLS equation with nonzero boundaries, including soliton solutions and analytical case distinctions.

Findings

Constructed and solved the Riemann-Hilbert problem.

Derived explicit dark, bright, and dark-bright soliton solutions.

Analyzed four distinct cases based on symmetry and boundary conditions.

Abstract

In this paper, the inverse scattering transform for the integrable discrete nonlocal PT symmetric nonlinear Schr\"odinger equation with nonzero boundary conditions is presented. According to the two different signs of symmetry reduction and two different values of the phase difference between plus and minus infinity, we discuss four cases with significant differences about analytical regions, symmetry, asymptotic behavior and the presence or absence of discrete eigenvalues, namely, the existence or absence of soliton solutions. Therefore, in all cases, we study the direct scattering and inverse scattering problem, separately. The Riemann-Hilbert problem is constructed and solved as well as the reconstruction formula of potential is derived, respectively. Finally, combining the time evolution, we provide the dark, bright, dark-bright soliton solutions on the nonzero background in difference cases under the reflectionless condition.