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

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.