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. Plentiful examples of WT are found in oceans, atmospheres, plasmas and Bose-Einstein condensates [1,2,3,4,5,6,7]. Roughly, there have been three major approaches to derive the WT theory, one based on a diagrammatic approach [8,10,9], the second based on cumulant expansions [2,4,11,7] and the third one, the random phase approximation (RPA) [1,3,5].

The diagrammatic approach was developed in a field theoretical spirit based on the Wyld's technique [8]. This method introduces an artificial Gaussian forcing for which a zero limit is taken at the end of the derivation. It is usually said that the statistical properties of this force (Gaussianity) do not affect the statistical properties of the resulting WT state which will be determined by the nonlinear properties only. However, such independence of the WT state on the statistics of the ``seed'' forcing is not obvious because the limit of small nonlinearity is taken before the limit of small force, i.e. the force remains much greater than the nonlinearity. In particular, when the nonlinearity parameter is strictly zero, the Wyld technique gives a Gaussian steady state which is clearly an artefact of this method because for linear systems statistics of the wave amplitudes remain the same as in the initial condition and, therefore, can be arbitrary. The question if any nonlinearity, no matter how small, can break this dependence of the steady state on the initial conditions still has not been answered in the literature. Thus, the diagrammatic approach, although a very efficient way to build the perturbation expansion, needs to be expanded to include non-Gaussian ``seed'' force in order to see to what extent the results are not sensitive to the force statistics. However, some elements of the Wyld technique will be used in the present paper, not as a complete description but rather as an auxiliary aid in writing out complicated terms.

The cumulant expansion approach differs from the other methods by working directly with the continuous Fourier transforms corresponding to the infinite coordinate space without introducing a finite box as an intermediate step. The main idea here is that, although the Fourier transform is ill-defined for the wave fields corresponding to homogeneous turbulence, it is well defined for the cumulants provided that the correlations decay rapidly enough in the coordinate space. The cumulant method is very elegant for describing the spectra and the multiple-point moments the points in which are not ``fused'' (i.e. all different). However, some important statistical quantities involve fused moments and they are hard (if at all possible) to define without introducing a finite box as an intermediate step. For example, one of such objects, $\langle \vert a\vert^4 \rangle$, is important because it describes intensity of fluctuations of the $k$-space distribution of energy $\vert a\vert^2$, namely $\delta E = \sqrt{\langle \vert a\vert^4 \rangle - \langle \vert a\vert^2 \rangle^2}$ (see [19]). Furthermore, non-decaying in the $x$-space correlations tend to naturally develop over the nonlinear time [19] and it is not clear what wave fields these correlations correspond to within the cumulant approach.

RPA approach has been by far most popular technique due to its clear intuitive content. However, this approach has occasionally been downgraded to just a convenient way of interpreting the results of a more rigorous technique based on the cumulant expansions. It happened because RPA, being widely used by physicists, had not been formulated rigorously. In particular, it is typically assumed 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 [17]. On the other hand, for three-wave systems (considered in this paper) 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 (not considered here) 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 [18,20] 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 such preservation was assumed but not proven. The goal of this paper will be to study the extent to which such an assumption is valid.

Another goal of this paper is to make RPA formulation more consistent by taking into account that both phases and the amplitudes are random variables. Indeed, even if one starts with a wavefield which has random phases but deterministic amplitudes, as it is typically done in numerical simulations, the amplitudes will get randomised because the nonlinear term producing their evolution contains (random) phase factors. Preliminary steps were recently done in [19,20] where we assumed that all the phases and the amplitudes in the initial wavefield are random variables independent of each other and that the phase factors are uniformly distributed on the unit circle on the complex plane. We kept the same acronym RPA but re-interpreted it as ``Random Phases and Amplitudes'', reflecting the fact that, first, the amplitudes are also random and, second, that it is not an ``approximation'' but rather an assumed property of the initial field. Such a generalised RPA was used in [19] to study the evolution of the higher moments of the Fourier amplitudes and in [20] to study their ``one-mode'' PDF. In fact, this form of RPA is more general than the cumulant approach because it can handle fields with long correlation lengths which appear to be important for intermittency [20].

Of course, for such an analysis to be trustworthy one should prove that the RPA properties hold over the nonlinear time and not just for the initial fields. Such mathematical validation of the RPA method will be in the focus of the present paper. To do this we will have to study the full joint PDF which involves the complete statistical information about the system, including the multi-mode correlations of both the amplitudes and the phase factors. We will derive an evolution equation for such PDF and we will show that it is identical to the equation obtained for the excitations in anharmonic crystals originally obtained by Peierls [15] and later reproduced by Brout and Prigogine [16] and Zaslavski and Sagdeev [17]. All these works were restricted to considering a quite narrow class of interaction Hamiltonians arising from a potential energy, i.e. depending on the coordinates but not momenta. These class does not include a large number of interesting WT systems, e.g. the capillary, internal, Rossby and Alfven waves. It is remarkable, therefore, that the Peierls equation turns out to be universal for the general class of three-wave systems, as it is shown in the present paper. Further, we use this equation to validate an ``essential'' RPA formulation, i.e. approximate RPA which holds only up to a certain order in nonlinearity and discreteness, but which is sufficient for the WT closure. This validation gives RPA technique a status of a rigorous approach which, due to the simplicity of its premises, is a winning tool for the future theory of non-Gaussianity of WT, its intermittency and interactions with coherent structures.

In addition to the mathematical validation of RPA, we will also develop WT further by considering new statistically important quantities. For a long time, describing and predicting the energy spectra was the only concern in WT theory. Recently, we presented a description of the higher order statistics of the one-point Fourier correlators in terms of their moments and PDF's. They describe the k-space ``noise'', i.e. the fluctuations of the mode energy about its mean value given by the energy spectrum. We also showed PDF's have a long algebraic tail which indicates presence of intermittency in WT fields. The present paper deals with phases, and we will therefore introduce and study some new correlators which will allow to describe the phase statistics directly. Such a description will compliment the mathematical validation of the RPA because it yields to a physical answer on how initially correlated phases can get de-correlated in the first place.