On linear instability of solitary waves for the nonlinear Dirac equation

Comech, Andrew; Guan, Meijiao; Gustafson, Stephen

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 639-654.

Nous considérons l'équation de Dirac non linéaire, aussi connue comme modèle de Soler. Nous étudions le spectre ponctuel des linéarisations autour d'ondes solitaires de petite amplitude dans la limite $\omega \to m$, et montrons que si $k>2/n$ une valeur propre positive et une négative sont présentes dans le spectre des linéarisations autour de ces ondes solitaires lorsque ω est suffisamment proche de m, ce qui entraîne que ces ondes solitaires sont linéairement instables. L'approche est basée sur l'application de la théorie des perturbations de Rayleigh–Schrödinger à la limite non relativiste de l'équation. Les résultats sont en accord formel avec le critère de stabilité de Vakhitov–Kolokolov.

We consider the nonlinear Dirac equation, also known as the Soler model:

$$i{\partial }_t\psi =-i\alpha ·\nabla \psi +m\beta \psi -{\left({\psi }^{⁎}\beta \psi \right)}^k\beta \psi ,m>0,\psi (x,t)\in {ℂ}^N,x\in {ℝ}^n,k\in \mathbb{N}.$$

We study the point spectrum of linearizations at small amplitude 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 ω sufficiently close to m, so that these solitary waves 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.

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