Darboux transformation and analytic solutions of the discrete \PT-symmetric nonlocal nonlinear Schrodinger equation

TL;DR

This paper develops a Darboux transformation method for the discrete PT-symmetric nonlocal nonlinear Schrödinger equation, enabling the derivation of various analytic solutions and analyzing their interactions.

Contribution

It introduces a novel algebraic iterative Darboux transformation for this specific nonlocal equation, expanding the toolkit for solving and understanding its solutions.

Findings

Derived breathing-soliton, periodic-wave, and rational soliton solutions.

Analyzed interactions between discrete rational dark and antidark solitons.

Provided asymptotic analysis of soliton interactions.

Abstract

In this letter, for the discrete parity-time-symmetric nonlocal nonlinear Schr\"{o}dinger equation, we construct the Darboux transformation, which provides an algebraic iterative algorithm to obtain a series of analytic solutions from a known one. To illustrate, the breathing-soliton solutions, periodic-wave solutions and localized rational soliton solutions are derived with the zero and plane-wave solutions as the seeds. The properties of those solutions are also discussed, and particularly the asymptotic analysis reveals all possible cases of the interaction between the discrete rational dark and antidark solitons.