On spectral stability of one- and bi-frequency solitary waves in Soler model in (3+1)D

TL;DR

This paper analyzes the spectral stability of one- and bi-frequency solitary waves in the 3D Soler model, using spherical harmonic decomposition to reduce the problem and demonstrating potential stability for bi-frequency solutions.

Contribution

It introduces a radial reduction technique for spectral stability analysis of solitary waves in the nonlinear Dirac Soler model, including bi-frequency solutions, which was not previously established.

Findings

Radial reduction simplifies spectral stability analysis.

Bi-frequency solitary waves can have stability properties similar to one-frequency waves.

The method applies spherical harmonic decomposition to the linearized operator.

Abstract

For the nonlinear Dirac equation with scalar self-interaction (the Soler model) in three spatial dimensions, we consider the linearization at solitary wave solutions and find the invariant spaces which correspond to different spherical harmonics, thus achieving the radial reduction of the spectral stability analysis. We apply the same technique to the bi-frequency solitary waves (which are generically present in the Soler model) and show that they can also possess linear stability properties similar to those of one-frequency solitary waves.