On stability of standing waves of nonlinear Dirac equations

TL;DR

This paper investigates the stability of standing waves in nonlinear Dirac equations, establishing conditions for orbital and asymptotic stability despite challenges posed by the indefinite energy spectrum.

Contribution

It provides the first stability results for nonlinear Dirac standing waves under specific spectral restrictions, extending stability analysis beyond Schrödinger models.

Findings

Proved orbital and asymptotic stability under certain spectral conditions

Identified limitations due to the indefinite energy spectrum

Extended stability concepts to nonlinear Dirac equations

Abstract

We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schr\"odinger equation, because of the strong indefiniteness of the energy.