English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

An effective equation describes a weakly nonlinear wave field evolutiongoverned by nonlinear dispersive PDEs \emph{via} the set of its resonances inan arbitrary big but finite domain in the Fourier space. We consider theSchr\"{o}dinger equation with quadratic nonlinearity including small externalrandom forcing/dissipation.  An effective equation is deduced explicitly for each case of monomialquadratic nonlinearities $ u^2, \, \bar{u}u, \, \bar{u}^2$ and the sets ofresonance clusters are studied. In particular, we demonstrate that thenonlinearity $\bar{u}^2$ generates no 3-wave resonances and its effectiveequation is degenerate while in two other cases the sets of resonances are notempty. Possible implications for wave turbulence theory are briefly discussed.

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