A theory for the spatial development of linearly unstable, coupled waves is presented in which both quasilinear and mode-coupling effects are treated in a self-consistent manner. Steady-state excitation of two waves (with frequencies ω1, ω2) is assumed at the boundary x=0, the plasma being homogeneous in the y and z directions. Coupled equations are derived for the x dependence of the amplitudes of the primary waves (ω1, ω2) and the secondary waves, nω1+mω2 (n and m being integers), correct through terms of second order in the wave amplitude, eφ/Te, but without the usual approximation of small growth rates. This general formalism is then applied to the case of coupled ion-acoustic waves driven unstable by an ion beam streaming in the direction of the x axis. If the modifications of the ion beam by the waves (“quasilinear” effects) are ignored, “explosive” instabilities (singularities in all of the amplitudes at finite x) are found even when all of the waves have positive energy. If these wave-particle interactions are included, the solutions are no longer singular, and all of the amplitudes have finite maxima, at locations in reasonable agreement with experimental results of Taylor and Ikezi.
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Burton D. Fried (1972) studied this question.