On linear instability of solitary waves for the nonlinear Dirac equation

Abstract

Abstract. We consider the nonlinear Dirac equation, also known as the Soler model:We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit ω → 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.Résumé. Nous considérons l'équation de Dirac non linéaire, aussi connu comme le modèle de Soler. Nous étudions le spectre ponctuel des linéarisations aux ondes solitaires des petites amplitudes dans la limite ω → m, et montrons que si k > 2/n, ensuite une valeur propre positive et une négative sont présents dans le spectre des linéarisations à ces ondes solitaires lorsque ω est suffisamment proche de m, ensuite ces ondes solitaires sont linéairement instable. L'approche est basée sur l'application de théorie de la perturbation 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.

Nonlinear Dirac equation solitary waves under a spinor force with different components

“…This result is consistent with the rigorous analysis in the nonrelativistic limit [42]. The spectral analysis of the NLD equation was also recently studied [43].…”

Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

“…In the nonrelativistic limit ω m, for k ∈ (0, 2), one has spectral stability according to [BC12b]; for k > 2, there is linear instability by [CGG14].…”

“…In view of recent results on stability and instability for the nonlinear Dirac equation [CKMS10,BC12a,BC12b,CGG14] it is becoming clear that the VK criterion is still useful for the spinor systems in the nonrelativistic limit, when the amplitude of solitary waves is small. In particular, the ground states ("smallest energy solitary waves") in the charge-subcritical nonlinear Dirac equation (with the nonlinearity of order 2k + 1, with k < 2/n) are linearly stable in the nonrelativistic limit ω m, which corresponds to solitary waves of small amplitudes.…”

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

“…This approach seemed to violate the continuity argument that the nonlinear Dirac (NLD) equation becomes a modified nonlinear Schrödinger (NLS) equation when ω approaches the mass parameter m of the Dirac equation from below. This argument has been made more rigorous by Comech [14]. Comech (private communication) has been able to prove that for κ < 2 the Vakhitov-Kolokolov [15] criterion guarantees linear stability in the nonrelativistic regime of the NLD equation for solutions of the form (in the rest frame) (x,t) = ψ(x)e −iωt where ω is less than but approximately equal to m. He was also able to show linear instability in the same nonrelativistic regime for κ > 2.…”