Explicit form of the effective evolution equation for the randomly forced Schr\"{o}dinger equation with quadratic nonlinearity

Abstract

An effective equation describes a weakly nonlinear wave field evolution governed by nonlinear dispersive PDEs \emph{via} the set of its resonances in an arbitrary big but finite domain in the Fourier space. We consider the Schr\"{o}dinger equation with quadratic nonlinearity including small external random forcing/dissipation. An effective equation is deduced explicitly for each case of monomial quadratic nonlinearities $ u^2, \, \bar{u}u, \, \bar{u}^2$ and the sets of resonance clusters are studied. In particular, we demonstrate that the nonlinearity $\bar{u}^2$ generates no 3-wave resonances and its effective equation is degenerate while in two other cases the sets of resonances are not empty. Possible implications for wave turbulence theory are briefly discussed.