lished. Experiments show that the gradual development of the wave continuum occurs due to the spectral broad- ening. Similar effects have been previously observed in liquid Helium [41] and in spin waves in ferrites [33]. However, the form of the resulting KZ-spectrum does not allow to conclude automatically whether the energy cascade is due to three- or four-wave interactions, and both explanations can be found in the literature. A di- rect way to clarify the matter would be to estimate the periods of the excited modes for the case of three- and four-wave interactions at the corresponding time scale. The experimental confirmation can then be obtained by measuring the corresponding wave fields [42]. irrotational flows) can be established by introducing a generalized velocity potential. In particular there is a Hamiltonian formulation in terms of two scalar variables, one of which is the free surface elevation [51]. In contrast to the three-dimensional waves, it is convenient to regard the y -axis as the vertical axis, with the waves propagating in the x -direction. The problem can be written in terms of a generalized velocity potential, satisfying with φ$_{x}$ = u + Ω y and φ$_{y}$ = v , [51]. In this coordinate system the free surface oscillates about the mean water level y = 0 . Introducing the new variable ζ , with A rotational water wave is a wave in which the un- derlying fluid flow exhibits non-zero vorticity. Physi- cally rotational waves describe wave-current interactions: a uniform current is described by zero vorticity (irrota- tional flow) [5, 7, 15], while a linearly sheared current has constant non-zero vorticity [18]. The irrotational flow setting is appropriate for waves generated by a distant storm and entering a region of water in uniform flow [34], while tidal flows are modelled by constant vortic- ity [16]. The existence of periodic gravity wave trains with small and large amplitudes [12, 14] was established, and qualitative properties of such waves were studied [6, 10, 11, 13, 14, 21, 23, 48]. In the case of capillary- or gravity-capillary waves the rigorous theory is limited to the small-amplitude regime [52], although some par- tial results pertaining to the existence of large-amplitude waves have recently been obtained [53]. Note that al- ready the small-amplitude theory is more involved in the presence of surface tension, including such phenomena as Wilton ripples [56] in which two harmonics with the ratio 1 : 2 are in resonance. An important new result on rotational capillary waves is the theorem on the dimension of flows with constant vorticity [8]: Theorem . Capillary wave trains can propagate at the free surface of a layer of water with a flat bed in a flow of constant non-zero vorticity only if the flow is two- dimensional. This theorem allows us to consider for flows of constant vorticity only two-dimensional flows propagating in the x -direction. For irrotational flows the existence of a ve- locity potential enables the transformation of the govern- ing equations for capillary water waves to a Hamiltonian system expressed solely in terms of the free surface and of the restriction of the velocity potential to the free sur- face [60]. The absence of a velocity potential for non-zero vorticities complicates the analysis considerably and one can not expect results of this type. However, for flows of constant vorticity the remarkable preservation of the main features of the above theory (for where In the new variables we have that linearizes the quadratic Hamiltonian and transforms Hamilton’s equations into with for k ∈ Z $^{∗}$. The change of variables Passing to Fourier variables can be seen as a change of variables and this transforms Hamilton’s equations into where the notation δF/δu is used for the variational derivative of a functional F with respect to the variable u . where ξ = φ | y = η is the restriction of the generalized ve- locity potential to the free surface, allows to obtain a canonical Hamiltonian system in the form [51]: IV. FLOWS WITH CONSTANT VORTICITY { Δ φ = 0 , −∞ < y < η, |∇ φ | → 0 , y → −∞ , (25) ζ = ξ − Ω 2 ∂ − 1 x η, (26) ˙ η = δH δξ , ˙ ξ = − δH δη , (27) ˙ η$_{k}$ = δH δξ ∗ k , ˙ ξ$_{k}$ = − δH δη ∗ k , (28) η$_{k}$ = √ | k | 2˜ ω ( k ) ( a$_{k}$ + a ∗ − $_{k}$) , ζ$_{k}$ = − i √ ˜ ω ( k ) 2 | k | ( a$_{k}$ − a ∗ − $_{k}$) ˜ ω ( k ) = √ σ | k | 3 + Ω 2 4 , (29) a$_{k}$ + i δH δa ∗ k = 0 . (30) H$_{2}$ = ∑ k ∈ Z ∗ ω ( k ) | a$_{k}$ | $^{2}$, (31) ω ( k ) = − Ω 2 sgn k + √ σ | k | 3 + Ω 2 4 , (32) 8