AMPLITUDE EQUATIONS FOR COUPLED ELECTROSTATIC-WAVES IN THE LIMIT OF WEAK INSTABILITY

We consider the simplest instabilities involving multiple unstable ele ctrostatic plasma waves corresponding to four-dimensional systems of m ode amplitude equations. In each case; the coupled amplitude equations are derived up to third-order terms. The nonlinear coefficients are s ingular in the limit in which the linear growth rates vanish together. These singularities are analysed using techniques developed in previo us studies of a single unstable wave. In addition to the singularities familiar from the single-mode problem, there are new singularities in coefficients coupling the modes. The new singularities are most sever e when the two waves have the same linear phase velocity and satisfy t he spatial resonance condition k(2) = 2k(1). As a result, the short-wa ve mode saturates at a dramatically smaller amplitude than that predic ted for the weak-growth-rate regime on the basis of single-mode theory . In contrast, the long-wave mode retains the single-mode scaling. If these resonance conditions are not satisfied then both modes retain th eir single-mode scaling and saturate at comparable amplitudes.