Explicit analytical solution of the nonlinear Vlasov-Poisson system

Abstract

In order to describe the time evolution of an inhomogeneous collisionless plasma, the nonlinear Vlasov equation is solved perturbatively, using the subdynamics approach and the diagrammatic techniques. The solution is given in terms of a double perturbation series: one with respect to the nonlinearities and the other with respect to the interaction between particles. The infinite sum of interaction terms can be performed exactly due to the property of dynamical factorization. Following the methodology, the exact solution in each order with respect to nonlinearities is computed. For a choice of initial perturbation the first-order exact solution is numerically integrated in order to find the local density excess. The approximate analytical solution is found to be in excellent agreement with exact numerical integration as well as with ab initio numerical simulations. Analytical computation gives a better insight into the problem, and it has the advantage to be simpler, and also accessible, in some range of parameters where it is difficult to find numerical solutions. © 1994 American Institute of Physics.