Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov–Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov–Kolokolov condition $dQ\mathrm{/}d\omega =0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross–Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and supercritical cases, showing the agreement with the Vakhitov–Kolokolov and the energy vanishing conditions.