On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential

TL;DR

This paper demonstrates that small standing waves in a nonlinear Dirac equation with a trapping potential act as attractors for small solutions, extending known results from the nonlinear Schrödinger equation to the Dirac setting.

Contribution

It establishes the stability and attractor properties of small standing waves in the nonlinear Dirac equation with a trapping potential, a novel extension from NLS results.

Findings

Small standing waves are attractors for small solutions.

Extension of NLS stability results to the Dirac equation.

Demonstrates asymptotic stability in the nonlinear Dirac setting.

Abstract

We consider a Dirac operator with short range potential and with eigenvalues. We add a nonlinear term and we show that the small standing waves of the corresponding nonlinear Dirac equation (NLD) are attractors for small solutions of the NLD. This extends to the NLD results already known for the Nonlinear Schr\"odinger Equation (NLS)