On linear instability of solitary waves for the nonlinear Dirac equation

We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi$, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp 2(\R)$, where $\alpha_j$, $j = 1,...,n$, and $\beta$ are $N \times N$ Hermitian matrices which satisfy $\alpha_j^2=\beta^2=I_N$, $\alpha_j \beta+\beta \alpha_j=0$, $\alpha_j \alpha_k + \alpha_k \alpha_j =2 \delta_{jk} I_N$. We study the spectral stability of solitary wave solutions $\phi(x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$, 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--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion.