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

Abstract

We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) 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 ±2ω eigenvalues of the linearization at a solitary wave, Bogoliubov 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.