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 ϕω(x)e−iωt to the nonlinear Dirac equation in Rn, for all n≥1, with the nonlinear term given by f(ψ⁎βψ)βψ (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 ω→m−0, in the case when f∈C1(R∖{0}), f(τ)=|τ|κ+O(|τ|K) for τ→0, with 0<κ<K. For n≥1, we prove the spectral stability of small amplitude solitary waves (ω⪅m) for the charge-subcritical cases κ⪅2/n (in particular, 1<κ≤2 when n=1) and for the “charge-critical case” κ=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 ±2mi. 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).