4-wave dynamics in kinetic wave turbulence

AgeneralHamiltonianwavesystemwithquarticresonancesisconsidered,inthestandardkineticlimitofacontinuumofweaklyinteractingdispersivewaveswithrandomphases.TheevolutionequationforthemultimodecharacteristicfunctionZisobtainedwithinan‘‘interactionrepresentation’’andaperturbationexpansion in the small nonlinearity parameter. A frequency renormalization is performed to removelinear terms that do not appear in the 3-wave case. Feynman–Wyld diagrams are used to average overphases,leadingtoafirstorderdifferentialevolutionequationforZ.Ahierarchyofequations,analogousto the Boltzmann hierarchy for low density gases is derived, which preserves in time the property ofrandomphasesandamplitudes.ThisamountstoageneralformalismforboththeN-modeandthe1-modePDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigatingintermittency.Someofthemainresultswhicharedevelopedhereindetailhavebeentestednumericallyinarecentwork