# Nonintegrable Spatial Discrete Nonlocal Nonlinear Schr\"odinger Equation

## TL;DR

This paper investigates a nonintegrable discrete nonlocal nonlinear Schrödinger equation, presenting numerical stationary solutions, analyzing their stability, and exploring the Cauchy problem to reveal new properties distinct from classical NLS solutions.

## Contribution

It introduces a nonintegrable discrete nonlocal NLS model, provides numerical solutions and stability analysis, and compares its behavior with classical NLS equations.

## Key findings

- Numerical stationary solutions are obtained using discrete Fourier transform.
- Linear stability of solutions is analyzed.
- Distinct properties are observed in the Cauchy problem solutions compared to classical NLS.

## Abstract

Integrable and nonintegrable discrete nonlinear Schr\"odinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time and reverse space-time nonlocal integrable equations, including nonlocal NLS, nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. And, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS which is direct discretization version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for NLS equation.