Nonlinear evolution of perturbations in marginally stable plasmas

We derive a general, self-consistent, reduced equation that describes the nonlinear evolution of electrostatic perturbations in marginally stable plasma equilibria. The equation is universal in the sense that it is independent of the equilibrium, and it contains as special cases the beam-plasma, the bump-on-tail, and the two-stream instability problems, among others. In particular, the present work offers a systematic justification of the O'Neil-Winfrey-Malmberg single-wave beam-plasma model. But more importantly, the analysis shows that the single-wave model has a wider range of applicability: it can be applied to localized perturbation in any marginally stable equilibrium. We discuss the linear theory, and construct families of exact nonlinear solutions. (C) 1998 Elsevier Science B.V.