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 (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.

17 pages

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Cited by in corpus (1)

• Solutions with compact time spectrum to nonlinear Klein--Gordon and Schroedinger equations and the Titchmarsh theorem for partial convolution