Parametric instability in continous media is the exponential growth of wave pairs with opposite wave vectors (if the pumping is uniform in space). The standard nonlinear theory of the instability considers the most important interaction of pairs of the waves. This interaction leads to renormalization of the pumping and finally to saturation of the instability on a certain level proportional to the pumping intensity. The corresponding theory (S-theory) is developed in detail.
The purpose of this talk is to show that in some cases S-theory describes only the intermediate asymptotics, and the final state is quite different. It can be demonstrated in the framework of a very simple model - the Nonlinear Schrödinger equation with defocusing nonlinearity and additional pumping and damping terms. In this case the final result of the instability is the formation of a "condensate" - a uniform oscillation of the medium. The final state is degenerate - the phase of the condensate can take two opposite values, and a final state can include a "kink" separating regions of different phases. It is remarkable that in the limit of very small pumping the level of the condensate remains finite - depending only on the frequency of the pumping.
The model described can be applied to antiferromagnets. A parametric pumping can be used for the generation of monochromatic signals of a very high intensity in nonlinear media of different kinds.