Exact time-dependent solutions of the Vlasov–Poisson equations

Abstract

The Vlasov–Poisson equations are relevant to collisionless plasmas and to stellar dynamics. They can be solved in one spatial dimension for an interesting class of cases by using a recent result about exact invariants of the motion of a particle in a one-dimensional potential. Lewis and Leach have given the necessary and sufficient conditions on the potential energy V(x,t) that an invariant which is quadratic in the momentum exist. For such a V(x,t), they exhibit the invariant explicitly. This result can be used to find the solutions of the one-dimensional Vlasov–Poisson equations for which the distribution functions are functions of quadratic functions of the momenta. A special case is the class of locally Maxwellian time-dependent solutions. The solutions for a single-species plasma, or a multispecies plasma where the charge to mass ratios are all equal, can be obtained by translating stationary solutions of the Vlasov–Poisson equations rigidly with an arbitrarily time-dependent displacement. If the charge to mass ratios of a multispecies plasma are unequal, then the solutions can be obtained by translating stationary solutions of modified Vlasov–Poisson equations with a displacement that depends quadratically on time. The modified Vlasov–Poisson equations include a species-dependent pseudogravity or ponderomotive force. This technique can be extended to obtain solutions of the Vlasov–Poisson or Vlasov–Maxwell equations in three dimensions.