Time-asymptotic wave propagation in collisionless plasmas.

We report the results of a new, systematic study of nonlinear longitudinal wave propagation in a collisionless plasma. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation, and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly, based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species, and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For "linearly stable" initial Vlasov equilibria, we obtain a critical initial amplitude (or threshold), separating the initial conditions that produce Landau damping to zero (A [triple bond] 0) from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping (A not identical with 0). These theoretical results have important implications about the stability of spatially uniform plasma equilibria, and they also explain why large-scale numerical simulations in some cases lead to zero-field final states whereas in others they yield nonzero multiple-traveling-wave final states.