Solution of the Linearized Vlasov Equation for a Space-Time-Dependent Distribution Function

Abstract

A solution is given for the one-dimensional linearized Vlasov equation when the background distribution function f0 is a sufficiently weak function of space and time; the field is given self-consistently through the Poisson equation and the initial value problem is explicitly considered. An eikonal solution, common to geometrical optics, is found, and the appropriate combination of such solutions so as to conform to given initial conditions is specified. Results are given for the modification of the Landau damping through the variation of f0, as well as for the coefficient of the eikonal exponential, which change is directly due to the space-time variation of f0. An example is given for an eikonal wave, in which wave-particle energy transfer is initially small, which is “driven” through the change of f0 to a condition where the solution breaks down due either to strong wave-particle effects or to the invalidity of the basic eikonal ordering assumption. Estimates for the times at which these limiting effects occur are given.