Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity

TL;DR

This paper analyzes the nonrelativistic limit of solitary wave solutions in the nonlinear Dirac equation with Soler-type nonlinearity, providing asymptotics crucial for stability analysis and revealing conditions for spectral degeneracy.

Contribution

It develops perturbation-based asymptotics for solitary waves in the nonlinear Dirac equation with non-differentiable nonlinearities, advancing understanding of their stability properties.

Findings

Asymptotic behavior of solitary waves in the nonrelativistic limit

Conditions for the absence of zero eigenvalue degeneracy

Implications for linear stability analysis

Abstract

We use the perturbation theory to build solitary wave solutions $\phi_\omega(x)e^{-i\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\bar\psi\psi)\beta\psi$, with $f(\tau)=|\tau|^k+o(|\tau|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $\omega\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schr\"odinger charge-critical, one has $Q'(\omega)<0$ for $\omega\lesssim m$, implying the absence of the degeneracy of zero eigenvalue of the linearization at a solitary wave.