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This paper addresses the linear and nonlinear three-dimensional propagationof an electron wave in a collisionless plasma that may be inhomogeneous,nonstationary, anisotropic and even weakly magnetized. The wave amplitude,together with any hydrodynamic quantity characterizing the plasma (density,temperature,...) are supposed to vary very little within one wavelength or onewave period. Hence, the geometrical optics limit is assumed, and the wavepropagation is described by a first order differential equation. This equationexplicitly accounts for three-dimensional effects, plasma inhomogeneity, Landaudamping, and the collisionless dissipation and electron acceleration due totrapping. It is derived by mixing results obtained from a direct resolution ofthe Vlasov-Poisson system and from a variational formalism involving a nonlocalLagrangian density. In a one-dimensional situation, abrupt transitions arepredicted in the coefficients of the wave equation. They occur when the stateof the electron plasma wave changes, from a linear wave to a wave with trappedelectrons. In a three dimensional geometry, the transitions are smoother,especially as regards the nonlinear Landau damping rate, for which a verysimple effective and accurate analytic expression is provided.

physics of plasmas

Envelope equation for the linear and nonlinear propagation of an  electron plasma wave, including the effects of Landau damping, trapping,  plasma inhomogeneity, and the change in the state of wave