Stability of solitary waves and vortices in a 2D nonlinear Dirac model

TL;DR

This paper analyzes the stability of solitary waves and vortices in a 2D nonlinear Dirac model, revealing conditions for neutral stability and instability, and comparing these findings with the nonlinear Schrödinger equation.

Contribution

It provides the first systematic spectral stability analysis of solitary waves and vortices in a 2D nonlinear Dirac model, highlighting stability regimes and vortex splitting behaviors.

Findings

Spinor solutions with a soliton and vortex can be neutrally stable over a wide frequency range.

Higher vorticity solutions are generally unstable and split into lower charge vortices.

The stability and dynamics differ significantly from the nonlinear Schrödinger equation.

Abstract

We explore a prototypical two-dimensional model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions consisting of a soliton in one component and a vortex in the other to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These results pave the way for a systematic stability and dynamics analysis of higher dimensional waveforms in a broad class of nonlinear Dirac models and a comparison revealing nontrivial differences with respect to their better understood non-relativistic analogue, the nonlinear Schr\"odinger equation.