content_image/424577.jpg content_image/424589.jpg content_image/424574.jpg content_image/424578.jpg content_image/424579.jpg content_image/424575.jpg content_image/424576.jpg content_image/424590.jpg content_image/424584.jpg
numerical simulations with two-triad clusters (fixed con- nection type, various coupling coefficients) can be found in [2]. 

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

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where 

and 

while ϕ$_{0}$ is the initial dynamical phase and the constants µ, K ( µ ) , τ are known explicitly as functions of I$_{13}$, I$_{23}$ and H$_{T}$ . 

The time evolution of a generic cluster again does not depend on the individual phases but on the dynamical phases; the corresponding equations for the dynamical phases have a form similar to the last equation of (17). For instance, in case of two triads a and b connected via 

one PP -connection (the NR -diagram is shown in Fig.5), the equations for the dynamical phases 

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read 

where the joint mode is denoted by C$_{1}$ and H is the Hamiltonian 

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the amplitudes has been studied numerically in [3]. It turned out that if initially ϕ$_{in}$ = 0 for a triad, then it stays zero and the changing range of amplitudes is max- imal. If ϕ$_{in}$ is very small but non-zero, say, ϕ$_{in}$ = 0 . 01 , 

## B. Evolution of the dynamical phases 

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then this range is substantially diminished: depending on the initial energy distribution within a triad it can de- crease 10 times and more, being minimal for ϕ$_{in}$ = π/ 2 . For the cluster shown in Fig.5 the maximal range of the amplitudes is observed for ϕ$_{a,in}$ = ϕ$_{b,in}$ = 0 . 

An interesting phenomenon has been uncovered in the case when at least one of the phases ϕ$_{a}$ and ϕ$_{b}$ is initially non-zero: a new time-scale is clearly observable where the amplitudes of the resonant modes are modulated by the evolution of the dynamical phases. The characteristic form of evolution is shown in Fig.6 for ϕ$_{a}$ = ϕ$_{b}$ = π/ 2 , [3]. 

This is a manifestation of one more crucial distinction between discrete and statistical wave turbulent regimes: while in the first regime phases are locked, in the latter they are supposed to be random. The effect of phase randomization among three-wave interactions of capillary waves has been studied recently experimentally in [40]. 

The transition from a coherent-phase to a random- phase system occurs above some excitation threshold when coherent wave harmonics broaden spectrally. The waves were excited parametrically in the range of the shaker frequencies 40 ÷ 3500 Hz and the modulation in- stability of capillary waves, which can be approximated by the squared secant function sech $^{2}$, has been estab- 

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