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

TL;DR

This paper investigates the linear stability of solitary waves in self-interacting spinor fields, establishing conditions based on the Vakhitov--Kolokolov criterion and energy vanishing, supported by numerical analysis in specific models.

Contribution

It generalizes the Vakhitov--Kolokolov approach to spinor fields and links real eigenvalue bifurcations to energy and charge conditions, with numerical validation.

Findings

Bifurcation of eigenvalues characterized by $dQ/d\,\omega=0$ and energy vanishing.

Numerical data confirms stability conditions in generalized models.

Agreement between theoretical conditions and numerical results.

Abstract

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 $dQ/d\omega=0$ 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.