Abstract
The nonlinear interaction between four electrostatic waves in a plasma is studied using the well defined phase approximation. For the quadruple process where the frequencies ω and wave-vectors k of the interacting waves satisfy the synchronism conditions ω1 + ω2 = ω3 + ω4 ; k1 + k2 = k3 + k4 it is shown that a necessary condition for the interaction to be explosively unstable is that s1 = s2 = - s3 = - s4 where sn = sgn (∂ ϵ/∂ ωn) and ϵ is the dielectric function. It is also shown that for four-wave coupling, the above criterion is not sufficient for an explosive instability. Solutions of the coupled-mode equations for the explosive regime are discussed and a growth rate and threshold for the instability are found. The theory is applied to the interaction between positive and negative energy electrostatic ion waves in a beam plasma system.
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