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

TL;DR

This paper derives explicit effective evolution equations for a weakly nonlinear Schrödinger equation with quadratic nonlinearity and random forcing, analyzing resonance structures and implications for wave turbulence.

Contribution

It provides explicit forms of effective equations for different quadratic nonlinearities, including resonance analysis and degeneracy cases, advancing understanding of wave turbulence models.

Findings

The effective equation for ar{u}^2 is degenerate with no 3-wave resonances.

Resonance sets are non-empty for u^2 and ar{u}u nonlinearities.

Implications for wave turbulence theory are discussed.

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.