On the spectral stability of the nonlinear Dirac equation of Soler type

Abstract

We study the spectral stability of the solitary wave solutions to the nonlin- ear Dirac equation in (1+1) dimension. We focus on a Soler type nonlinear model, where the nonlinearity is given by (ψψ)^p. The method we use con- sists in perturbe the solutions with a sufficiently small function ρ, finding a time evolution equation for this perturbation where this equation depends on the spectrum of the linearized operator Hμ. We will say that the solitary wave solutions are stable if the spectrum of Hμ does not have eigenvalues with imaginary part other than zero. We were only able to provide bounds for the real and imaginary part of the discrete spectrum of Hμ. In the end, we summarize what is known about σ(Hμ).