Nonlinear Landau Damping

In this article, we report the mathematical details of a systematic analysis of the nonlinear Landau damping of longitudinal electrostatic waves propagating in a collisionless plasma. Many of the main results have been reported previously; unfortunately, major parts of the essential mathematical developments had to be omitted for various reasons, making it almost impossible for even the most well-prepared reader to follow the analysis. Sufficient details are provided here to remedy this situation. Some important results that have not been reported previously are also are included here. Most notably among these is the distinction between strong branches and weak branches of nonzero time-asymptotic electric field amplitudes that bifurcate from the zero-amplitude solution for the time-asymptotic electric field, and the results on the weak branches that had to be omitted in the earlier report of this research. 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, separating the initial conditions that Landau damp to zero from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping. These theoretical results explain why in some cases experiments and large-scale numerical simulations have resulted in zero-field final states; whereas in other cases they have yielded nonzero multiple-traveling-wave final states because the theoretical results establish the existence of a “threshold” in the initial electric field below which the field damps to zero and above which it evolves to a finite-amplitude multiple-traveling-wave final state.