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

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: $i\partial_tψ=-iα\partial_xψ+mβψ-f(ψ^\astβψ)βψ$, $ψ(x,t)\in\C^2$, $x\in\R$, $f\in C^\infty(\R)$, $m>0$, where $α$, $β$ are $2\times 2$ hermitian matrices which satisfy $α^2=β^2=1$, $αβ+βα=0$. We study the spectral stability of solitary wave solutions $ϕ_ω(x)e^{-iωt}$. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at solitary waves of arbitrarily small amplitude, in the limit $ω\to m$. We prove that if $f(s)=s^k+O(s^{k+1})$, $k\in\N$, with $k\ge 3$, then one positive and one negative eigenvalue are present in the spectrum of linearizations at all solitary waves with $ω$ 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-Schrödinger 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).