Space and time behavior of parametric instabilities for a finite pump wave duration in a bounded plasma

The space and time behavior of the decay waves is computed analytically in the regime of standard parametric decay. The plasma is assumed to be homogeneous and bounded. The pump wave has a finite pulse duration. The propagation of the pump wave is taken into account, its depletion is ignored. The parametric growth is solved in terms of fluctuating initial and boundary conditions corresponding to thermal noise at equilibrium. Fluctuating source terms, representing noise emission, are accordingly retained in the coupled mode equations. The initial stage of parametric growth is investigated in detail; the time from which the asymptotic concept of absolute or convective instability applies is computed. The connection between the Maniey-Rowe and flux conservation relations is discussed.