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 ±k 0. 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 k 0 mode of the electric field is dominated by the Landau solution with (negative) damping decrement γ L . The third‐order k 0 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 k 0 and ε, γ L < γ N . Thus at sufficiently long times, the modified third‐order mode dominates the solution. For certain ε and k 0 this behavior resembles the results of Knorr and Armstrong, obtained by numerical integration of the nonlinear Vlasov equation.