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arXiv · 2006.03345

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

Nabile Boussaid·Claudio Cacciapuoti·Raffaele Carlone·Andrew Comech·Diego Noja·Andrea Posilicano
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Abstract

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 $\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetry. We show that a perturbation which breaks the $\mathbf{SU}(1,1)$-symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a perturbation which breaks both the $\mathbf{SU}(1,1)$-symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. The developing instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $\pm 2\omega\mathrm{i}$.

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Nabile Boussaid, Claudio Cacciapuoti, Raffaele Carlone, Andrew Comech, Diego Noja, Andrea Posilicano. 2020-06-05. Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity. https://arxiv.org/abs/2006.03345

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