Resonance clustering in wave turbulent regimes: Integrable dynamics

Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in k-space, and 2) existence of “gaps” in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: kinetic, described by wave kinetic equations and power energy spectra; discrete, characterized by resonance clustering; and 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. Integrable clusters yield pattern formation in kinetic and mesoscopic wave turbulent regimes. PACS: 10.180,-a, 67.40.Hf, 68.03Kn, 68.03Hj