Existence of Travelling Wave Solutions to the Maxwell-Pauli and Maxwell-Schr\"odinger Systems

TL;DR

This paper proves the existence of travelling wave solutions in Maxwell-Schr"odinger and Maxwell-Pauli systems, showing their energies behave like classical kinetic energy at small velocities, indicating an effective mass equal to the bare mass.

Contribution

It establishes the existence of travelling wave solutions for both Maxwell-Schr"odinger and Maxwell-Pauli models with detailed energy behavior analysis.

Findings

Travelling wave solutions exist for both models at moderate speeds.

Energy of solutions approximates classical kinetic energy at low velocities.

Effective mass matches the bare mass of the particle.

Abstract

We study two mathematical descriptions of a charged particle interacting with it's self-generated electromagnetic field. The first model is the one-body Maxwell-Schr\"odinger system where the interaction of the spin with the magnetic field is neglected and the second model is the related one-body Maxwell-Pauli system where the spin-field interaction is included. We prove that there exist travelling wave solutions to both of these systems provided that the speed $|v|$ of the wave is not too large. Moreover, we observe that the energies of these solutions behave like $\frac{mv^2}{2}$ for small velocities of the particle, which may be interpreted as saying that the effective mass of the particle is the same as it's bare mass.