# Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

## TL;DR

This paper investigates solitary waves in a two-dimensional nonlinear Dirac equation, analyzing their existence, stability, and evolution from discrete to continuum models in waveguide arrays.

## Contribution

It demonstrates the continuation of continuum Dirac solitons across discretization levels and explores their stability and dynamics in discrete and continuum regimes.

## Key findings

- Continuum Dirac solitons persist for all discretization parameters.
- Solutions with 1- or 2-sites at the anti-continuum limit can be continued to large couplings.
- Some solutions are found to be spectrally unstable and exhibit specific dynamical behaviors.

## Abstract

In the present work, we explore a nonlinear Dirac equation motivated as the continuum limit of a binary waveguide array model. We approach the problem both from a near-continuum perspective as well as from a highly discrete one. Starting from the former, we see that the continuum Dirac solitons can be continued for all values of the discretization (coupling) parameter, down to the uncoupled (so-called anti-continuum) limit where they result in a 9-site configuration. We also consider configurations with 1- or 2-sites at the anti-continuum limit and continue them to large couplings, finding that they also persist. For all the obtained solutions, we examine not only the existence, but also the spectral stability through a linearization analysis and finally consider prototypical examples of the dynamics for a selected number of cases for which the solutions are found to be unstable.