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modes have been identified while in [20] the development of full broadband spectra is observed. 

It is tempting to use the parameter 

as a characteristic allowing to distinguish between dif- ferent wave turbulent regimes: taking a sufficiently large number of modes one will get the classical kinetic regime. However, the situation is more complicated. As it was shown in [57], Fig.4, low-frequency excitation (beginning with 25 Hz) generates distinct peaks of wave frequen- cies while increasing the excitation to 300 Hz yields an isotropic turbulent regime with a power law distribution of energies. Further increasing the excitation frequency generates an unexplained peak at 400 Hz. Moreover, cap- illary waves in Helium do notice the geometry of the ex- perimental facilities for L/λ ∼ 30 − 300 , the correspond- ing dispersion in circular tanks being described by Bessel functions (G. Kolmakov, private communication). In this case there exist nonlinear resonances and their manifes- tation has been recently established in laboratory exper- iments with capillary waves on the surface of superfluid Helium, [1]. The local maximum of the wave amplitudes is detected at frequencies of the order of the viscous cut- off frequency in the case when the surface is driven by a periodical low-frequency force. This means that the en- ergy is concentrated in a few discrete modes at the very end of the inertial interval ( k$_{1}$, k$_{2}$ ) , in the beginning of the dissipative range. 

These novel results are paramount both from theoret- ical and practical point of view. This is the first exper- imental confirmation of the predictions of the model of laminated wave turbulence that resonant modes can ex- ist within and outside the inertial interval in the large k -scales. 

The origin of these peaks (similar to rogue waves in the oceans) in the wave spectrum can be seen from the Fig.2: whether or not a rogue wave is generated depends both on the driving frequency drive and on the resonant frequencies in a wave system. Indeed, suppose the modes M$_{1}$, M$_{2}$, M$_{3}$ have frequencies $_{1}$,$_{2}$ ,$_{3}$ and form a resonance 1 + 2 = $_{3}$. If drive ≈ 3 (the mode with high frequency which is unstable), then the appearance of peaks with frequencies 1 and 2 is to be expected. If the modes M$_{2}$, M$_{3}$, M$_{4}$ form a resonance with 2 = 3 + $_{4}$, then an excitation with drive ≈ 2 yields peaks at 3 and $_{4}$. The importance of the high-frequency mode is discussed in the next section. 

Notice that rogue waves can appear within and out- side the inertial interval while discrete wave turbulent regimes are not connected with the existence of the in- ertial interval. The situation is quite general and does not depend on the type of waves under consideration. This is in agreement with the experimental observations 

Characteristic wavelengths 

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in superfluid Helium where rogue waves have been de- tected and the fact that the energy balance is nonlocal in k -space was established [19]. 

The usual way to characterize the physical meaning of the resonance width δ is to regard it as a shift in the resonant wave frequencies, cf. [40, 47, 61], yielding reso- nance broadening in a wide wave spectrum with a large number of modes. 

To characterize the broadening in a three-wave system governed by (12) one can choose as an appropriate pa- rameter the inverse nonlinear oscillation time of resonant modes 

In discrete regimes the phases of individual modes are co- herent; however if the width | Δ$_{,B}$$_{3}$ | of the high-frequency mode in a resonant triad becomes substantially larger that τ − 1 k , the coherence is lost — this being a necessary condition for the occurrence of the kinetic regime. One concludes then that 

• The discrete regime corresponds to 

• The kinetic regime corresponds to 

• The mesoscopic regime corresponds to 

## Characteristic resonance broadening 

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