Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

Abstract

We study the point spectrum of the linearization at a solitary wave solution ${\varphi }_{\omega }(x)e^{-\mathrm{i}\omega t}$ to the nonlinear Dirac equation in ${\mathbb{R}}^n$, $n\ge 1$, 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 nonzero real part, in the non-relativistic limit $\omega \lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus \{0\})$, $f(\tau )=|\tau {|}^k+O(|\tau {|}^K)$ for $\tau \to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($\omega \lesssim m$) for the charge-subcritical cases $k\lesssim 2/n$ ($1<k\le 2$ when $n=1$) and for the "charge-critical case" $k=2/n$, $K>4/n$. An important part of the stability analysis is the proof of the absence of bifurcations of nonzero-real-part eigenvalues from the embedded threshold points at $\pm 2m\mathrm{i}$. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model, using this family to determine the multiplicity of $\pm 2\omega \mathrm{i}$ eigenvalues of the linearized operator, and the analysis of the behaviour of "nonlinear eigenvalues" (characteristic roots of holomorphic operator-valued functions).

arXiv comment55 pages