On linear instability of solitary waves for the nonlinear Dirac equation

Abstract

We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb tψ=-iα\cdot \nabla ψ+m βψ-f(ψ\sp\ast βψ) βψ$, $\psi (x,t)\in {\mathbb{C}}^N$, $x\in {\mathbb{R}}^n$, $n\le 3$, $f\in C\sp 2(\R)$, where ${\alpha }_j$, $j=1,...,n$, and $\beta$ are $N\times N$ 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 $\varphi (x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega \to m$, proving that if $k>2/n$, 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.

17 pages. arXiv admin note: substantial text overlap with arXiv:1203.3859 (an earlier 1D version)

References in corpus (3)

• Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity
• On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation
• Solitary Wave Solutions for the Nonlinear Dirac Equations