Introduction

Imagine surface waves on sea produced by wind of moderate strength, so that the surface is smooth and there is no whitecaps. Typically, these waves exhibit great deal of randomness and the theory which aims to describe their statistical properties is called Wave Turbulence (WT). More broadly, WT deals the fields of dispersive waves which are engaged in stochastic weakly nonlinear interactions over a wide range of scales in various physical media. Plentiful examples of WT are found in oceans, atmospheres, plasmas and Bose-Einstein condensates [1,2,3,4,5,6,7,10,11,12,30]. WT theory has a long and exciting history which started in 1929 from the pioneering paper of Peierls who derived a kinetic equation for phonons in solids [19]. In the 1960's these ideas have been vigorously developed in oceanography [6,5,2,4,30] and in plasma physics [3,11,9]. First of all, both the ocean and the plasmas can support great many types of dispersive propagating waves, and these waves play key role in turbulent transport phenomena, particularly the wind-wave friction in oceans and the anomalous diffusion and thermo-conductivity in tokamaks. Thus, WT kinetic equations where developed and analysed for different types of such waves. A great development in the general WT theory was done by in the papers of Zakharov and Filonenko [5]. Before this work it was generally understood that the nonlinear dispersive wavefields are statistical, but it was also thought that such a ``gas'' of stochastic waves is close to thermodynamic equilibrium. Zakharov and Filonenko [5] were the first to argue that the stochastic wavefields are more like Kolmogorov turbulence which is determined by the rate at which energy cascades through scales rather than by a thermodynamic ``temperature'' describing the energy equipartition in the scale space. This picture was substantiated by a remarkable discovery of an exact solution to the wave-kinetic equation which describes such Kolmogorov energy cascade. These solutions are now commonly known as Kolmogorov-Zakharov (KZ) spectra and they form the nucleus of the WT theory.

Discovery of the KZ spectra was so powerful that it dominated the WT theory for decades thereafter. Such spectra were found for a large variety of physical situations, from quantum to astrophysical applications, and a great effort was put in their numerical and experimental verification. For a long time, studies of spectra dominated WT literature. A detailed account of these works was given in [1] which is the only book so far written on this subject. Work on these lines has continued till now and KZ spectra were found in new applications, particularly in astrophysics [16], ocean interior [18] and even cosmology [27]. However, the spectra do not tell the whole story about the turbulence statistics. In particular, they do not tell us if the wavefield statistics is Gaussian or not and, if not, in what way. This question is of general importance in the field of Turbulence because it is related with the intermittency phenomenon, - an anomalously high probability of large fluctuations. Such ``bursts'' of turbulent wavefields were predicted based on a scaling analysis in [32] and they were linked to formation of coherent structures, such as whitecaps on sea [28] or collapses in optical turbulence [7]. To study these problems qualitatively, the kinetic equation description is not sufficient and one has to deal directly with the probability density functions (PDF).

In fact, such a description in terms of the PDF appeared already in the the Peierls 1929 paper simultaneously with the kinetic equation for waves [19]. This result was largely forgotten by the WT community because fine statistical details and intermittency had not interested turbulence researchers until relatively recently and also because, perhaps, this result got in the shade of the KZ spectrum discovery. However, this line of investigation was continued by Brout and Prigogine [20] who derived an evolution equation for the multi-mode PDF commonly known as the Brout-Prigogine equation. This approach was applied to the study of randomness underlying the WT closures by Zaslavski and Sagdeev [21]. All of these authors, Peierls, Brout and Prigogine and Zaslavski and Sagdeev restricted their consideration to the nonlinear interaction arising from the potential energy only (i.e. the interaction Hamiltonian involves coordinates but not momenta). This restriction leaves out the capillary water waves, Alfven, internal and Rossby waves, as well as many other interesting WT systems. Recently, this restriction was removed by considering the most general three-wave Hamiltonian systems [24]. It was shown that the multi-mode PDF still obeys the Peierls-Brout-Prigogine (PBP) equation in this general case. This work will be described in the present review. We will also present, for the first time, a derivation of the evolution equation for the multi-mode PDF for the general case of four wave-systems. This equation is applicable, for example, to WT of the deep water surface gravity waves and waves in Bose-Einstein condensates or optical media described by the nonlinear Schroedinger (NLS) equation.

We will also describe the analysis of papers [24] of the randomness assumptions underlying the statistical WT closures. Previous analyses in this field examined validity of the random phase assumption [20,21] without devoting much attention to the amplitude statistics. Such ``asymmetry'' arised from a common mis-conception that the phases evolve much faster than amplitudes in the system of nonlinear dispersive waves and, therefore, the averaging may be made over the phases only ``forgetting'' that the amplitudes are statistical quantities too (see e.g. [1]). This statement become less obvious if one takes into account that we are talking not about the linear phases $\omega t$ but about the phases of the Fourier modes in the interaction representation. Thus, it has to be the nonlinear frequency correction that helps randomising the phases [21]. On the other hand, for three-wave systems the period associated with the nonlinear frequency correction is of the same $\epsilon^2$ order in small nonlinearity $\epsilon$ as the nonlinear evolution time and, therefore, phase randomisation cannot occur faster that the nonlinear evolution of the amplitudes. One could hope that the situation is better for 4-wave systems because the nonlinear frequency correction is still $\sim \epsilon^2$ but the nonlinear evolution appears only in the $\epsilon^4$ order. However, in order to make the asymptotic analysis consistent, such $\epsilon^2$ correction has to be removed from the interaction-representation amplitudes and the remaining phase and amplitude evolutions are, again, at the same time scale (now $1/\epsilon^4$). This picture is confirmed by the numerical simulations of the 4-wave systems [26,23] which indicate that the nonlinear phase evolves at the same timescale as the amplitude. Thus, to proceed theoretically one has to start with phases which are already random (or almost random) and hope that this randomness is preserved over the nonlinear evolution time. In most of the previous literature prior to [24] such preservation was assumed but not proven. Below, we will describe the analysis of the extent to which such an assumption is valid made in [24].

We will also describe the results of [22] who derived the time evolution equation for higher-order moments of the Fourier amplitude, and its application to description of statistical wavefields with long correlations and associated ``noisiness'' of the energy spectra characteristic to typical laboratory and numerical experiments. We will also describe the results of [23] about the time evolution of the one-mode PDF and their consequences for the intermittency of stochastic nonlinear wavefields. In particular, we will discuss the relation between intermittency and the probability fluxes in the amplitude space.