Nonlocal reductions of the Ablowitz-Ladik equation

TL;DR

This paper develops the inverse scattering transform for the nonlocal PT-symmetric Ablowitz-Ladik equation, deriving soliton solutions and spectral properties, thus extending integrable systems theory to nonlocal discrete nonlinear Schrödinger equations.

Contribution

It introduces the inverse scattering framework for the nonlocal Ablowitz-Ladik equation, including spectral analysis, soliton solutions, and the interpretation of the transform as a generalized Fourier transform.

Findings

Derived 1- and 2-soliton solutions

Proved the completeness relation for Jost solutions

Established the inverse scattering transform as a generalized Fourier transform

Abstract

The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and the fundamental analytic solutions. In addition, the paper studies the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness relation for the associated Jost solutions is proved. Based on this, the expansion formula over the complete set of Jost solutions is derived. This will allow one to interpret the inverse scattering transform as a generalised Fourier transform.