Nonlinear Behaviors of Waves in a Plasma in the Vicinity of Marginally Stable State

Nonlinear behaviors of monochromatic traveling waves are discussed by the use of an asymptotic method. The wave is assumed to be stationary in the first approximation and nonlinear interactions are taken into account only among the wave under consideration and its harmonics. It is shown that two cases are distinguished according as the frequency of the wave in the first approximation is given by a simple root or a double root of the dispersion relation. Characteristic behaviors of the wave in the latter case are investigated by discussing a model equation and special cases of two-beam instability. The behaviors of the complex amplitude of the wave are visualized in the form of “a motion of a particle in an axisymmetric potential field”. Various phenomena, such as amplitude oscillation, explosion of amplitude, amplitude-dependent frequency shift, are found to take place depending on the plasma and the initial condition.