Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity

Details Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity

2012-03-17

arxiv.org/abs/1203.3859v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: $i\partial_t\psi=-i\alpha\partial_x\psi+g(\psi^\ast\beta\psi)\beta\psi$, $\psi(x,t)\in\C^2$, $x\in\R$; $g\in C^\infty(\R)$, $m:=g(0)>0$, where $\alpha$, $\beta$ are $2\times 2$ hermitian matrices which satisfy $\alpha^2=\beta^2=1$, $\alpha\beta+\beta\alpha=0$. 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 solitary waves of arbitrarily small amplitude, in the limit $\omega\to m$. We prove that if $g(s)=m-s^k+O(s^{k+1})$, $k\in\N$, with $m>0$ and $k\ge 3$, then one positive and one negative eigenvalue are present in the spectrum of linearizations at all solitary waves with $\omega$ sufficiently close to $m$. This shows that all solitary waves of sufficiently small amplitude 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. Let us mention a similar independent result [Guan-Gustafson] on linear instability for the nonlinear Dirac equation in three dimensions, with cubic nonlinearity (this result is also in formal agreement with the Vakhitov-Kolokolov stability criterion).

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