Commun. Comput. Phys., 10 (2011), pp. 1211-1240.


Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

Miguel D. Bustamante 1, Elena Kartashova 2*

1 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.
2 IFA, J. Kepler University, Linz 4040, Austria.

Received 11 September 2010; Accepted (in revised version) 16 February 2011
Available online 2 August 2011
doi:10.4208/cicp.110910.160211a

Abstract

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 review paper we present the results on 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 show 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.


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PACS: 47.10.Df, 47.10.Fg, 02.70.Dh
Key words: Resonance clustering, Hamiltonian formulation, wave turbulent regimes, NR-diagram, integrable dynamics.

*Corresponding author.
Email: miguel.bustamante@ucd.ie (M. D. Bustamante), elena.kartaschova@jku.at (E. Kartashova)
 

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