Exact, time-dependent solutions of the one-dimensional Vlasov–Maxwell equations

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Abstract
Coordinate transformations from a reference frame with electrostatic Bernstein–Greene–Kruskal equilibria produce exact, time‐dependent one‐particle distribution functions and electric fields that obey the nonlinear one‐dimensional Vlasov–Maxwell equations. The Liouville equation invariant on which the distribution function depends is the transformed particle energy. Procedures for deriving invariants of more general momentum dependence than quadratic are suggested. The vanishing of the longitudinal current density is imposed upon the exact time‐dependent Vlasov–Poisson solutions and a particular case of a time oscillatory Bernstein–Greene–Kruskal equilibria is solved. The existence of solutions with a nonconstant time‐stretching factor ρ(t) is demonstrated.