A Model for the Maxwell Equations Coupled with Matter Based on Solitons

Abstract

We present a simple model of interaction of the Maxwell equations with a matter field defined by the Klein–Gordon equation. A simple linear interaction and a nonlinear perurbation produces solutions to the equations containing hylomorphic solitons, namely stable, solitary waves, whose existence is related to the ratio energy/charge. These solitons, at low energy, behave as poinwise charged particles in an electromagnetic field. The basic points are the following ones: (i) the matter field is described by the Nonlinear Klein–Gordon equation with a suitable nonlinear term; (ii) the interaction is not described by the equivariant derivative, but by a very simple coupling which preseves the invariance under the Poincaré group; (iii) the existence of soliton can be proved using the tecniques of nonlinear analysis and, in particular, the Mountain Pass Theorem; (iv) a suitable choice of the parameters produces solitons with a prescribed electric charge and mass/energy; (v) thanks to the point (ii), the dynamics of these solitons at low energies is the same of classical charged particles.