Discussion

In the present paper, we considered evolution of the full N-mode objects such as the generating functional and the probability density function for all the wave amplitudes and their phase factors. We proved that the phase factors, being statistically independent and uniform on $S^1$ initially, remain so over the nonlinear evolution time in the leading order in small nonlinearity. If in addition the initial amplitudes are independent too, then they remain so over the nonlinear time in a weak sense. Namely, all joint PDF's for the number of modes $M \ll N$ split into products of the one-mode densities with $O(M/N)$ and $O({\epsilon}^2)$ accuracy. Thus, the full $N$-mode PDF does not factorise as a product of $N$ one-mode densities and the Fourier modes in the set considered as a whole are not independent. However, the wave turbulence closure only deals with the joint objects of the finite size $M$ of variables while taking $N \to \infty$ limit. These objects do factorise into products and, for the WT purposes, the Fourier modes can be interpreted as statistically independent. In particular, the derivation of the kinetic equation for the energy spectrum deals only with the $1$-mode and the $2$-mode distributions and is, therefore, justified by the results of the present paper. Generally speaking, our results reduce the leading-order WT problem to the study of the one-mode amplitude PDF's and they validate the generalised RPA technique introduced in [19,20]. Such a study of the one-mode PDF and the high-order momenta of the wave amplitudes was done in [19,20]. It was shown, in particular, that anomalous probabilities of large wave amplitudes can appear in the form of a finite-flux solution in the amplitude space caused by a wave-breaking amplitude cutoff. The reader is referred to these papers for the discussion of the WT intermittency.

Although our results indicate that correlations between 2 or more (but $\ll N$) modes do not appear in the leading (i.e. ${\epsilon}^2$) order for the three-wave systems, they definitely appear as corrections in the next (i.e. ${\epsilon}^4$) order. Our paper is concerned with the main order statistics only in which the main evolution happens in the $1$-mode objects, e.g. the $1$-mode amplitude distributions. For study of the multi-mode correlations developing in WT in the next order in ${\epsilon}$ the reader is referred to papers [21,22].

We have also considered correlators of the phase and we showed the relation between the statistical properties of the phase $\phi$ and the phase factors $\psi = e^{i\phi}$. We showed that the mean of $\phi$ and its fluctuations about the mean grow in time and, therefore, there exist no $2 \pi$-wide interval in which the phase would remain uniformly distributed. Moreover, phases $\phi$ become correlated at different wavenumbers that lie on the resonant manifold. These properties make the phase $\phi$ an inappropriate variable for formulating the RPA method of WT description. On the other hand, our work shows that the phase factors $\psi = e^{i\phi}$ do remain statistically independent and uniform on $S^1$ which makes them the right choice for the RPA formulation.

The present paper deals with the three-wave systems only. The four-wave resonant interactions are slightly more complicated in that the nonlinear frequency shift occurs at a lower order in nonlinearity parameter than the nonlinear evolution of the wave amplitudes. To build a consistent description of the amplitude moments one has to perform a renormalisation of the perturbation series taking into account the nonlinear frequency shift. This derivation will be published separately, whereas here we just announce its main result, the 4-wave generalisation of the Peierls equation for the PDF. It has the same continuity equation form (36) but now the probability flux is

$$F_j = 4\pi{\epsilon}^4\sum_{123} W_{23}^{j1}\delta(\tilde\omega^{ji}_{2... ...ac{\delta}{\delta s_1} -\frac{\delta}{\delta s_2} -\frac{\delta}{\delta s_3})P,$$ (62)

where $W_{23}^{j1}$ is the 4-wave ineraction coefficient and $\tilde \omega^{l\alpha}_{\mu\nu} = \omega^{l\alpha}_{\mu\nu} +\Omega_l+\Omega_\alpha-\Omega_\mu-\Omega_\nu$ with $\Omega_l = 2 \epsilon \sum_\mu W^{l \mu}_{l \mu} n_\mu$ being the nonlinear frequency shift. As wee see this equation is even more compact than its 3-wave analog. In addition to the derivation of this equation, we will also analyse its properties and consequences for the mode correlations and intermittency in 4-wave turbulent systems.