Narrow Results

We consider the nonlinear Klein–Gordon–Maxwell system derived from the Lagrangian on four-dimensional Minkowski space-time, where ϕ is a complex scalar field and F_μν = ∂_μ𝔸_ν - ∂_ν𝔸_μ is the electromagnetic field. For appropriate nonlinear potentials , 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 𝕁^B, 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 , and δ = e^{1 - k} for as e → 0. We work entirely in the energy norm, and the approximation is controlled in this norm for times of .