Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

TL;DR

This paper investigates the spectral stability of solitary waves in a nonlinear Dirac equation with concentrated nonlinearity, analyzing how different symmetry-breaking perturbations affect their stability and leading to instability through eigenvalue bifurcations.

Contribution

It provides a detailed spectral analysis of solitary waves under symmetry-breaking perturbations in a nonlinear Dirac equation with concentrated nonlinearity, highlighting conditions for stability and instability.

Findings

Symmetry-breaking perturbations can preserve or destroy spectral stability.

Breaking both $ ext{SU}(1,1)$ and parity symmetries leads to instability.

Eigenvalue bifurcations cause instability in weakly relativistic solitary waves.

Abstract

We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetry. We show that a perturbation which breaks the $\mathbf{SU}(1,1)$-symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a perturbation which breaks both the $\mathbf{SU}(1,1)$-symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. The developing instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $\pm 2\omega\mathrm{i}$.