Andrew Comech, Meijiao Guan, Stephen Gustafson
We consider the nonlinear Dirac equation, also known as the Soler model: \[ i\p\sb t\psi=-i\bm\alpha\cdot\bm\nabla\psi+m\beta\psi-f(\psi\sp\ast\beta\psi)\beta\psi, \ \quad \psi(x,t)\in\C^{N}, \ \quad x\in\R^n, \ \quad f\in C\sp 2(\R), \] 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 $\phi(x)e^{-i\omega t}$. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at the small amplitude solitary waves in the limit $\omega\to m$. We consider dimensions n=1, 2, and 3, and prove that if $f(s)=s^k+O(s^{k+1})$, $k\in\N$, with $m>0$ and either $k\ge 3$, n=1, or $k\ge 2$, n=2, or k=1, n=3, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\omega$ sufficiently close to $\omega=m$. This shows in particular that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh-Schr\"odinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion.
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http://arxiv.org/abs/1209.1146
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