Resonance clustering in wave turbulent regimes: Integrable dynamics

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 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.

Discrete wave turbulence of rotational capillary water waves

“…with the coefficients α 1j , ..., α 5j being explicit functions of I 13 , I 23 and H T , for j = 1, 2, 3, [31]. The energy of each mode E j is proportional to the square of its amplitude, E j (t) ∼ C 2 j (t) ∼ dn 2 , i.e.…”

“…As one can see from (17), the time evolution of a resonant triad does not depend on the individual phases of the resonant modes but on the dynamical phase ϕ corresponding to the resonance conditions. In this case, the closed expression for the evolution of dynamical phase, first found in [31], reads…”

“…The system (43) is integrable for arbitrary initial conditions, if Z a /Z b is equal to 1, 2 or 1/2, [49]. In the case of arbitrary Z a /Z b , the integrability can be proven for some specific initial conditions [31].…”