Single-mode saturation of a linearly unstable plasma

The nonlinear oscillations of an electron plasma described by the collisionless Vlasov equation are studied using a perturbation technique previously applied by Simon and Rosenbluth [Phys. Fluids 19, 1567 (1976)]. It is proved by a characteristic argument that the plasma is globally stable, so that Bogoliuboff’s method of ‘‘secular regularization’’ is applicable. Assuming the plasma is confined in a box, and that only the lowest mode is unstable, it is shown that the ‘‘eigenmode dominance’’ approximation of Simon and Rosenbluth fails to conserve energy, but that energy and momentum conservation can be regained by considering interaction between the discrete and continuum modes. A formula is derived for the amplitude and phase of the saturated nonlinear oscillations. In a subsidiary result, it is shown that nonlinear effects damp the steady‐state oscillations predicted by linearized theory for some stable plasmas.