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$N \times N$Hermitianmatriceswhichsatisfy$\alpha_j^2=\beta^2=I_N$,$\alpha_j \beta+\beta \alpha_j=0$,$\alpha_j \alpha_k + \alpha_k \alpha_j =2 \delta_{jk} I_N$\mathrm{.}Westudythespectralstabilityofsolitarywavesolutions$\phi(x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$,provingthatif$k>2/n$, 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.