Abstract: We prove the spectral stability (absence of linear instability) in the nonlinear Dirac equation

with the scalar self-interaction, known as the Soler model.

It turns out that the linearization at a solitary wave can have eigenvalues with positive real part (the ones

that lead to instability) only near the origin and near the embedded thresholds (located at 2mi and -2mi).

We show that the absence of these "unstable" eigenvalues near the origin is guaranteed by the Vakhitov--

Kolokolov stability criterion from the NLS theory. We then find a sufficient condition for the absence of

such eigenvalues near embedded thresholds. It turns out that both conditions are satisfied for certain

subcritical powers (for example, for powers between cubic and quintic in one-dimensional case), thus

proving spectral stability in these cases.

The results are based on the construction of bi-frequency solitary waves and applying the Keldysh theory

of characteristic roots ("nonlinear eigenvalues").

This is a joint research with Nabile Boussaid, University Bourgogne -- Franche Comte (Besançon).

The results are presented in arXiv:1705.05481.