On linear instability of solitary waves for the nonlinear Dirac equation

We consider the nonlinear Dirac equation, also known as the Soler model: i partial derivative(t)psi = -i alpha . del psi + m beta psi - (psi* beta psi)(k) beta psi, m >0, psi (x,t)is an element of C-N, x is an element of R-n, k is an element of N. We study the point spectrum of linearizations at small amplitude solitary waves in the limit omega -> m, proving that if k > 2/n, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with omega sufficiently close to m, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh Schrodinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion. (c) 2013 Elsevier Masson SAS. All rights reserved.