On global attraction to solitons for 3D Maxwell-Lorentz equations

TL;DR

This paper proves that solutions of finite energy in the 3D Maxwell-Lorentz system with a rotating charge tend to soliton solutions over time, demonstrating a form of global attraction in the system.

Contribution

It establishes the long-time convergence of finite energy solutions to solitons in the Maxwell-Lorentz equations with a rotating charge, a novel result in this context.

Findings

Solutions converge to solitons in local energy norms

Long-time behavior is characterized by attraction to static field configurations

The result applies to finite energy initial conditions

Abstract

We consider the Maxwell field coupled to a single rotating charge. This Hamiltonian system admits soliton-type solutions, where the field is static, while the charge rotates with constant angular velocity. We prove that any solution of finite energy converges, in suitable local energy seminorms, to the corresponding soliton in the long time limit.