Results on the spectral stability of standing wave solutions of the Soler model in 1-D

TL;DR

This paper investigates the spectral stability of standing wave solutions in a 1D nonlinear Dirac equation (Soler model), providing bounds on eigenvalues and spectral characterizations for power nonlinearities.

Contribution

It offers new spectral bounds and stability criteria for standing waves in the 1D nonlinear Dirac equation with specific nonlinearities.

Findings

Eigenvalue bounds for linearized operators around standing waves

Spectral characterization of ground states based on inner product conditions

Identification of frequency ranges with no unstable eigenvalues

Abstract

We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(\langle\psi,\beta \psi\rangle_{\mathbb{C}^2}) \beta$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. For the case of power nonlinearities $f(s)= s |s|^{p-1}$, $p>0$, we obtain a range of frequencies $\omega$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langle\phi_0,\beta \phi_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schr\"odinger case.