Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics (Communications in Computational Physics, 2011)
Two fundamental facts of the modern wave turbulence theory are 1) existenceof power energy spectra in k-space, and 2) existence of "gaps" in this spectracorresponding to the resonance clustering. Accordingly, three wave turbulent regimesare singled out: kinetic, described by wave kinetic equations and power energy spectra;discrete, characterized by resonance clustering; and mesoscopic, where both typesof wave field time evolution coexist. In this review paper we present the results onintegrable dynamics of resonance clusters appearing in discrete and mesoscopic waveturbulent regimes. Using a novel method based on the notion of dynamical invariantwe show that some of the frequently met clusters are integrable in quadratures for arbitraryinitial conditions and some others-only for particular initial conditions. We alsoidentify chaotic behaviour in some cases. Physical implications of the results obtainedare discussed.