On the motion of charged particles in constant electromagnetic field: the parallel case

This paper is devoted to presenting a rigorous mathematical derivation for the classical phenomenon in Maxwell's theory that a charged particle moves along a straight line in a constant electromagnetic field if the initial velocity is parallel to the constant electromagnetic field. The particle is modeled by scaled solitons to a class of nonlinear Klein-Gordon equations and the nonlinear interaction between the charged particle and the electromagnetic field is governed by the Maxwell-Klein-Gordon system. We show that when the size and amplitude of the particle are sufficiently small, the solution to the coupled nonlinear system exists up to any given time and the energy of the particle concentrates along a straight line. The method relies on the modulation approach for the study of stability for solitons and weighted energy estimates for the Maxwell-Klein-Gordon equations.