Parametric instabilities induced by the coupling of high and low frequency plasma modes

A general dispersion relation is derived for parametric instabilities caused by a large amplitude pump wave which couples high and low frequency modes of a magnetized plasma. The nonlinear effects coupling the high and low frequency waves are: (1) the low-frequency pondermotive force due to the coupling of the pump with the high frequency perturbations, and (2) the high frequency nonlinear source current density produced by coupling of the pump with the low frequency perturbations. The dispersion relation contains decay instabilities, purely growing instabilities, modulational instabilities, self-focusing and nonlinear Landau damping. As examples the dispersion relation is applied to self-focusing, magnetosonic decay and purely growing instabilities, whistler decay and nonlinear Landau dumping of electron plasma waves and transverse electromagnetic waves.