Scattering of Solitons for Dirac Equation Coupled to a Particle

TL;DR

This paper proves that solutions to a Dirac equation coupled with a particle tend to a combination of a traveling wave and a free wave over time, using spectral theory and symplectic projection.

Contribution

It establishes soliton-like asymptotics for the coupled Dirac-particle system, demonstrating long-term convergence in energy norm.

Findings

Solutions with initial states near the solitary manifold converge to a traveling and free wave

Convergence occurs in the global energy norm

Spectral theory and symplectic projection are key tools used

Abstract

We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold in the Hilbert phase space.