Nonlinear saturation of two unstable modes; survival competition

Abstract

A bounded uniform one‐dimensional collisionless plasma is considered. The velocity distribution is of the ’’bump‐on‐tail’’ type and two linear eigenmodes with closely spaced wave vectors have phase velocities located on the lowest part of the positive slope of the bump. All other eigenmodes are assumed reasonably separated in phase velocity and located on negative (stable) slope regions. The time‐asymptotic saturated nonlinear solutions are evaluated by a recently developed variation of the Bogoliubov method. In the ’’quasi‐linear’’ approximation (i.e., zero‐frequency mode coupling terms, alone, are retained) the only stable solution is one in which the initially fastest growing mode attains its ’’single‐mode’’ amplitude and the second mode has zero amplitude. In the full nonlinear calculation, the result is the same unless the two modes are above a critical distance from the bottom of the slope. Above this critical value, there are two stable solutions. One as just described. The other in which the lower mode reaches its full single‐mode amplitude and the first has zero amplitude. No stable solution exists for which both modes have finite amplitude. Similar mode survival competition may obtain in multi‐mode situations.