Instabilities in a Plasma Coexisting with a Large-Amplitude Wave

The stability of a periodic large-amplitude electrostatic wave [Bernstein-Greene-Kurskal (BGK) mode] in a collisionless plasma is investigated. It is found that instability or damping can result whenever a wave has a frequency which is resonant with one of the natural frequencies of a particle in the BGK mode. The total growth or damping rate is expressed as a summation over these resonances. In appropriate limits, the growth rates are identical with those corresponding to linear and nonlinear Landau damping. Three specific problems are considered. Electron plasma waves in a plasma at thermal equilibrium are shown to be unstable toward decay into an ion acoustic wave. Plasma waves in an electron-beam plasma system are shown to be unstable toward decay into ion waves. Finally, an electron plasma wave at frequency ω₀ is shown to be unstable toward decay into the lower sideband, that is a wave at ω₀-ωT where ωT is the trapping frequency.