On spectral stability of solitary waves of nonlinear Dirac equation on a line

TL;DR

This paper investigates the spectral stability of solitary wave solutions to the one-dimensional nonlinear Dirac equation with cubic nonlinearity, providing numerical and analytical evidence that these waves are spectrally stable.

Contribution

It offers the first numerical computation of the spectrum of linearization at solitary waves in the nonlinear Dirac equation and derives explicit eigenfunctions, extending results to generic nonlinearities.

Findings

Solitary waves are spectrally stable according to numerical spectrum analysis.

Explicit eigenfunctions for the linearized operator are obtained.

Some analytical results apply broadly to nonlinear Dirac equations with different nonlinearities.

Abstract

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.