## 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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### References:

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