Critical Initial Amplitude of Langmuir Wave Damping

By one-dimensional Vlasov-Poisson simulation, the critical initial state marking the transition between the Landau scenario, in which the electric field is definitively damped to zero and the O’Neil scenario, in which the Landau damping is stopped after a certain damping stage, are studied. It is found that the critical initial amplitude ?* can only exist when the product of the wave number (km) and the electron thermal velocity (vth) is moderate, that is, 0.2