Effective dynamics for solitons in the nonlinear Klein Gordon Maxwell system and the Lorentz force law

TL;DR

This paper develops a rigorous perturbation theory for solitons in the nonlinear Klein-Gordon Maxwell system, showing their long-term stability and deriving their effective dynamics governed by the Lorentz force law.

Contribution

It introduces a new dynamical perturbation framework for gauge-invariant solitons in the Klein-Gordon Maxwell system and derives their effective equations of motion.

Findings

Solitons are long-time stable under external electromagnetic perturbations.

Effective dynamics of solitons follow the Lorentz force law.

The analysis is valid in the small electromagnetic coupling limit.

Abstract

We consider the nonlinear Klein Gordon Maxwell system on four dimensional Minkowski space-time. For appropriate nonlinearities the system admits soliton solutions which are gauge invariant generalizations of the non-topological solitons introduced and studied by T.D. Lee and collaborators for pure complex scalar fields. We develop a rigorous dynamical perturbation theory for these solitons in the small e limit, where e is the electromagnetic coupling constant. The main theorems assert the long time stability of the solitons with respect to perturbation by an external electromagnetic field produced by the background current, and compute their effective dynamics to O(e). The effective dynamical equation is the equation of motion for a relativistic particle acted on by the Lorentz force law familiar from classical electrodynamics.