Growth rates of parametric instabilities driven by two pumps

The parametric instability growth rate of ion-acoustic and Langmuir waves, driven unstable by two uniform pumps near the Bohm--Gross frequency, is calculated as a function of pump amplitudes and frequencies. Two instability mechanisms can be identified: one corresponds to the usual, single pump parametric instabilities (decay and oscillating two stream), while the other is similar to that found in the Mathieu equation. The interaction between these two mechanisms results in a nonmonotonic dependence of the growth rate on the pump amplitudes and frequencies: both cancellation and enhancement are obtained for various values of the parameters. An analytic study of the relevant dispersion relation using Hill's method is complemented by numerical studies in both the frequency and time domains