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

Abstract

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: $i{\partial }_t\psi =-i\alpha {\partial }_x\psi +m\beta \psi -f({\psi }^{\ast }\beta \psi )\beta \psi$, $ψ(x,t)\in\C^2$, $x\in \mathbb{R}$, $f\in C^{\infty }(\mathbb{R})$, $m>0$, 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 ${\varphi }_{\omega }(x)e^{-i\omega t}$. 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 $f(s)=s^k+O(s^{k+1})$, $k\in \mathbb{N}$, with $k\ge 3$, 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-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).

15 pages

References in corpus (1)

• Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity

Cited by in corpus (1)

• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry