On stability of standing waves of nonlinear Dirac equations

We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schr\"odinger equation, because of the strong indefiniteness of the energy.