Preservation of the RPA property.

Validity of the WT theory relies on persistence of the RPA property of the wavefield over the nonlinear evolution time. Such persistence was demonstrated in [5] based on the evolution equation for the multi-mode PDF, where the product factorization of PDF was shown to hold with an accuracy sufficient for the WT closure. It was also emphasized in [5] that RPA must use independent phase factors $\psi_k$ rather than the phases $\phi_k$ independence of which does not survive over the nonlinear time. The theoretical prediction of persistent independence of $A_k$'s and $\psi_k$'s and about the growth of correlations of $\phi_k$'s will be checked in this paper numerically.

Further, the WT approach predicts that the mean value of the phase grows with a rate given by the nonlinear frequency correction and that the r.m.s. fluctuation of the phase also grows in time [5]. In this paper we will see that in reality the time evolution of the phase is more complicated than this WT prediction: the phase exhibits quasi-periodic fluctuations intermittent by rare ``phase runs'', - monotonic changes over several linear periods by large values which can significantly exceed $2 \pi$.