Spectral stability of bi-frequency solitary waves in Soler and Dirac--Klein--Gordon models

TL;DR

This paper constructs bi-frequency solitary waves in nonlinear Dirac models, linking their spectral stability to that of single-frequency waves, with implications for quantum computing.

Contribution

It introduces bi-frequency solitary waves in Dirac models and relates their stability to known single-frequency wave stability, advancing understanding of their spectral properties.

Findings

Bi-frequency solitary waves are constructed in the Soler and Dirac--Klein--Gordon models.

Spectral stability of bi-frequency waves reduces to that of single-frequency waves.

The work connects eigenvalues, symmetry, and existence of these waves.

Abstract

We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction (the Soler model) and the Dirac--Klein--Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing. We show the relation of $\pm 2\omega\mathrm{i}$ eigenvalues of the linearization at a solitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.