On linear instability of solitary waves for the nonlinear Dirac equation

TL;DR

This paper analyzes the spectral stability of solitary waves in the nonlinear Dirac equation, demonstrating that certain solitary waves are linearly unstable when bifurcating from NLS solutions near the relativistic limit.

Contribution

It provides a rigorous spectral analysis showing the presence of unstable eigenvalues for solitary waves close to the nonrelativistic limit, extending stability criteria to the nonlinear Dirac equation.

Findings

Existence of positive and negative eigenvalues indicating instability

Unstable solitary waves bifurcate from NLS solutions as frequency approaches mass

Results align with the Vakhitov--Kolokolov stability criterion

Abstract

We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi$, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp 2(\R)$, 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 $\phi(x)e^{-i\omega t}$. 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.