A review of the nonlinear theory of plasma oscillations

Abstract

This is a coherent review of various aspects of nonlinear longitudinal plasma oscillations. The two cases of a single mode and of a continuum of modes are treated separately. In the monochromatic case, the important role of resonant particles is discussed. It is shown that secular terms arise when the Vlasov and Poisson equations are solved by a regular perturbation method in powers of the electric field. The application of singular perturbation methods to remove secularities is treated. In the second case the quasi-linear theory is reviewed and Dupree's extension of it is also discussed. After Al'tshul and Karpman's method of summing the principal secular terms to all orders, the kinetic equation of waves, wave-particle interactions and resonant wave interactions are discussed. Finally the plasma echo phenomenon is treated briefly.