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Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity. (English) Zbl 1426.35026

Summary: We study the point spectrum of the linearization at a solitary wave solution \(\phi_\omega(x) e^{- \operatorname{i} \omega t}\) to the nonlinear Dirac equation in \(\mathbb{R}^n\), for all \(n \geq 1\), with the nonlinear term given by \(f(\psi^\ast \beta \psi) \beta \psi\) (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit \(\omega \to m - 0\), in the case when \(f \in C^1(\mathbb{R} \smallsetminus \{0 \}), f(\tau) = | \tau |^\kappa + O(| \tau |^K)\) for \(\tau \to 0\), with \(0 < \kappa < K\). For \(n \geq 1\), we prove the spectral stability of small amplitude solitary waves \((\omega \lessapprox m)\) for the charge-subcritical cases \(\kappa \lessapprox 2 / n\) (in particular, \(1 < \kappa \leq 2\) when \(n = 1)\) and for the “charge-critical case” \(\kappa = 2 / n\) (with \(K > 4 / n)\).
An important part of the stability analysis is the proof of the absence of bifurcation of nonzero-real-part eigenvalues from the embedded threshold points at \(\pm 2 m \operatorname{i}\). Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model and on the analysis of the behavior of “nonlinear eigenvalues” (characteristic roots of holomorphic operator-valued functions).

MSC:

35B35 Stability in context of PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35C08 Soliton solutions
35P15 Estimates of eigenvalues in context of PDEs
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