Third-Order Oscillations in a Maxwellian Plasma

Abstract

Nonlinear plasma oscillations in a classical, nonrelativistic, collisionless, Maxwellian electron gas are considered. There is assumed a small, sinusoidal variation in the spatial part of the initial distribution function, corresponding to excitation of wavenumber modes ±k0. The nonlinear Vlasov equation is solved to third order in the long time limit via the Montgomery-Gorman perturbation expansion, where the expansion parameter is ε, the amplitude of the initial perturbation. The linear, first-order k0 mode of the electric field is dominated by the Landau solution with (negative) damping decrement γL. The third-order k0 mode is modified, however, by singling out the spatially uniform part of the distribution function for special treatment, much in the manner of the quasi-linear theory. A nonlinear damping decrement results such that, for many values of k0 and ε, γL < γN. Thus at sufficiently long times, the modified third-order mode dominates the solution. For certain ε and k0 this behavior resembles the results of Knorr and Armstrong, obtained by numerical integration of the nonlinear Vlasov equation.