Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models
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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 \begin{document}$± 2ω\mathrm{i}$\end{document} eigenvalues of the linearization at a solitary wave, Bogoliubov \begin{document}$\mathbf{SU}(1,1)$\end{document} 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.