Soliton solutions and their dynamics in reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr\"odinger equations

TL;DR

This paper introduces reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schrödinger equations, derives multi-soliton solutions, and explores their complex dynamics, including breathing and collapsing behaviors.

Contribution

It presents the first derivation of multi-soliton solutions for these nonlocal discrete DNLS equations using Hirota's method and analyzes their dynamic behaviors.

Findings

Solitons exhibit breathing and periodic collapse for certain parameters.

Solitons remain nonsingular within specific parameter ranges.

Rich soliton structures are revealed in the nonlocal equations.

Abstract

In this paper, we introduce the reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr\"odinger (DNLS) equations through the nonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation. The muti-soliton solutions of two types of nonlocal discrete derivative nonlinear Schr\"odinger equations are derived by means of the Hirota bilinear method and reduction approach. We also investigate the dynamics of soliton solutions and reveal the rich soliton structures in the reverse-space and reverse-space-time nonlocal discrete DNLS equations. Our investigation shows that the solitons of these nonlocal equations often breathe and periodically collapse for some soliton parameters, but remain nonsingular for other range of parameters.