The higher-dimensional Ablowitz-Ladik model: from (non-)integrability and solitary waves to surprising collapse properties and more exotic solutions

TL;DR

This paper investigates a two-dimensional Ablowitz-Ladik discretization of the nonlinear Schrödinger model, revealing non-integrability, stable solitary waves near the continuum limit, and exotic solutions like vortices and line solitons.

Contribution

It demonstrates the non-integrability of the 2D Ablowitz-Ladik model and explores its stable solitary waves and exotic solutions, contrasting with standard discretizations.

Findings

Near the continuum limit, the model exhibits no collapse and stable spectral waves.

The model admits exact line solitons and discrete vortices.

Singularity confinement suggests the discretization is non-integrable.

Abstract

We propose a consideration of the properties of the two-dimensional Ablowitz-Ladik discretization of the ubiquitous nonlinear Schrodinger (NLS) model. We use singularity confinement techniques to suggest that the relevant discretization should not be integrable. More importantly, we identify the prototypical solitary waves of the model and examine their stability, illustrating the remarkable feature that near the continuum limit, this discretization leads to the absence of collapse and complete spectral wave stability, in stark contrast to the standard discretization of the NLS. We also briefly touch upon the three-dimensional case and generalizations of our considerations therein, and also present some more exotic solutions of the model, such as exact line solitons and discrete vortices.