Exact time-dependent solutions of the Vlasov--Poisson equations

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