Resonance clustering in wave turbulent regimes: Integrable dynamics
arXiv:1002.4994
Abstract
Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in -space, and 2) existence of "gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic}, where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.
This is Work In Progress carried out in years 2008-2008 and partly supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the project SCIEnce, Contract No. 026133