  For the nonlinear Dirac equation in (1+1)D with scalar self-interaction
(Gross--Neveu model), with quintic and higher order nonlinearities (and within
certain range of the parameters), we prove that solitary wave solutions are
asymptotically stable in the "even" subspace of perturbations (to ignore
translations and eigenvalues $\pm 2\omega i$). The asymptotic stability is
proved for initial data in $H^1$. The approach is based on the spectral
information about the linearization at solitary waves which we justify by
numerical simulations. For the proof, we develop the spectral theory for the
linearized operators and obtain appropriate estimates in mixed Lebesgue spaces,
with and without weights.
