ABSTRACT
The stability analysis of collisionless plasma, electron plasma oscillations in particular, were based on a linearized model. In that description, amplitudes of unstable waves continue to grow exponentially, and the analysis eventually ceases to be valid when the assumption of linearization brakes down. Quasi-linear theory is one of the models which can answer that question, at least for some cases. Quasi-linear models for weakly dispersive waves are not well developed, although some weakly nonlinear models for wave-particle interactions have been proposed. Within the simple quasi-linear theory the fluctuations reach a stationary level since the growth is arrested and no damping is included in the analysis. Conservation of the average particle density demonstrated before ensures that the area under the distribution function is conserved; hence the final plateau is uniquely determined by the requirement that the two areas labeled 1 and 2 are equal.
