Particle paths and phase plane for time-dependent similarity solutions of the one-dimensional Vlasov-Maxwell equations

Abstract

Lie group point transformations applied to the one-dimensional Vlasov– Maxwell equations yield general similarity forms for the dependent and independent variables. One class of such solutions is seemingly like Bern-stein-Greene-Kruskal solutions in allowing a relatively free choice of electric field, but with a more complex time dependence. The phase trajectories of the particles are found here for both temporally damped and (possibly unphysical) growing electric fields in this class by numerical integration in the original and in transformed co-ordinates. The analysis, which includes an analytic consideration of phase-plane fixed (critical) points, shows the advantages of the new co-ordinates, and reveals qualitative features of the distribution function.