Time evolution equation for the Probability Distribution Function (PDF) is derived for system of weakly interacting waves. It is shown that a steady state for such system may correspond to strong intermittency.

Random Phase Approximation (RPA) provides a very convenient tool to study the ensembles of weakly interacting waves, commonly called wave turbulence. In its traditional formulation, RPA assumes that phases of interacting waves are random quantities but it usually ignores randomness of their amplitudes. Recently, RPA was generalised in a way that takes into account the amplitude randomness and it was applied to study of the higher momenta and probability densities of wave amplitudes. However, to have a meaningful description of wave turbulence, the RPA properties assumed for the initial fields must be proven to survive over the nonlinear evolution time, and such a proof is the main goal of the present paper. We derive an evolution equation for the full probability density function which contains the complete information about the joint statistics of all wave amplitudes and phases. We show that, for any initial statistics of the amplitudes, the phase factors remain statistically independent uniformly distributed variables. If in addition the initial amplitudes are also independent variables (but with arbitrary distributions) they will remain independent when considered in small sets which are much less than the total number of modes. However, if the size of a set is of order of the total number of modes then the joint probability density for this set is not factorisable into the product of one-mode probabilities. In the other words, the modes in such a set are involved in a "collective" (correlated) motion. We also study new type of correlators describing the phase statistics.

Time evolution equation for the Probability Distribution Function (PDF) is derived for system of weakly interacting waves. It is shown that a steady state for such system may correspond to strong intermittency.Introduction -Wave Turbulence (WT) is a common name for the fields of dispersive waves which are engaged in stochastic weakly nonlinear interactions over a wide range of scales. Numerous examples of WT are found in oceans, atmospheres, plasmas and Bose-Einstein condensates [1][2][3][4][5][6][7][8][9]. For a long time, describing and predicting the energy spectra was the only concern in WT theory. More recently, some attention was given to the study of turbulence intermittency. WT intermittency, or "burstiness" of the turbulent signal, was observed experimentally and numerically and was attributed, as in most turbulent systems, to the presence of coherent structures. Examples include collapsing filaments in Bose-Einstein condensates with attractive potentials [9,10], condensate quasi-solitons in systems with repulsive potentials [9,11,12], white caps of sea waves at small scales [13], freak ocean waves at larger scales [14]. Often, such coherent structures are intense but quite sparse so that in most of the space waves remain weakly nonlinear and mostly unaffected by these structures.