EFFECTIVE DYNAMICS FOR SOLITONS IN THE NONLINEAR KLEIN–GORDON–MAXWELL SYSTEM AND THE LORENTZ FORCE LAW

Abstract

We consider the nonlinear Klein–Gordon–Maxwell system derived from the Lagrangian [Formula: see text] on four-dimensional Minkowski space-time, where ϕ is a complex scalar field and Fμν = ∂μ𝔸ν - ∂ν𝔸μ is the electromagnetic field. For appropriate nonlinear potentials [Formula: see text], the system admits soliton solutions which are gauge invariant generalizations of the non-topological solitons introduced and studied by Lee and collaborators for pure complex scalar fields. In this article, 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 𝕁B, 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. The theorems are valid in a scaling regime in which the external electromagnetic fields are O(1), but vary slowly over space-time scales of [Formula: see text], and δ = e1 - k for [Formula: see text] as e → 0. We work entirely in the energy norm, and the approximation is controlled in this norm for times of [Formula: see text].