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

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 $\mathbf{S}\mathbf{U}(1,1)$ 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.