Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the SU(1, 1) symmetry: the first preserving and the second breaking the parity symmetry. We show that a particular perturbation which breaks the SU(1, 1) symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a particular perturbation which breaks both the SU(1, 1) symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. This instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues ±2ωi.