The theory of stochastic wavefields in weakly nonlinear dispersive media has a long and exciting history which started in 1929 when Peierls derived his kinetic equation for phonons in solids [1]. Applications of these ideas appeared in the physics of the ocean and atmosphere [2,3,4,5,6,7,8], laboratory and astrophysical plasmas [10,11,12], Bose condensates and nonlinear optics [14], anharmonic crystals [1,15,16]. Any attempt to give a fair historical review would be doomed in such a short letter and we refer an interested reader for further references to the book [17] and a more recent review [18]. The common name that has arisen for all these approaches is Wave Turbulence (WT).
WT closure requires, besides weak nonlinearity, randomness in both the phases and the amplitudes of the Fourier modes. Namely, all the phases and all the amplitudes must be statistically independent of each other, in some sense, and the phases must be uniformly distributed. Such an approach was recently formulated in [19,21] as a generalization of the Random Phase Approximation (RPA) much loved by the physicists which, in its traditional form, ignores the amplitude randomness [17]. We even kept the same acronym RPA but now read it as ``Random Phases and Amplitudes''. Below, in section 2.1, we define explicitly what we mean by RPA. RPA does not fix the shape of the probability densities of the individual mode amplitudes and, therefore, it allows one to consider wavefields with non-decaying correlations which is helpful because such long correlations tend to arise naturally in WT systems. In [19], we used RPA to describe the arbitrary-order moments of the wave amplitude, and in [21] we extended this approach to describing the one-mode probability density function (PDF) and considered solutions for this PDF corresponding to intermittency. In these works, however, RPA was assumed (but not proven) to hold over the nonlinear time.
Such a proof is the main goal of the present paper. We shall consider initial fields of the RPA type, and we will prove that the RPA properties are preserved (i.e. no phase or amplitude correlations are generated with accuracy sufficient for the WT closure) over the nonlinear evolution time. In order to do this we shall derive an evolution equation for the full multi-mode PDF which will turn out to be the Zaslavski-Sagdeev (ZS) equation [13] (a WT cousin of the Brout-Prigogine equation for anharmonic crystals [15,16]). We will show that, for any statistics of the amplitudes, phases tend to stay random if they were so initially. If, in addition, the initial amplitudes are independent variables they will remain independent in a coarse-grained sense, i.e. when considered in small subsets which are much smaller than the total set of modes.
The original paper by ZS [13] was also devoted to the study of the applicability of the WT closure and, therefore, it is appropriate here to mention in which way our approach is different. First, ZS consider 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. In our work we remove this restriction by considering the most general three-wave Hamiltonian equation (11) and we show that the multi-mode PDF still obeys the ZS equation in this case. Secondly, ZS studied the phase statistics only, whereas our work considers both the phases and the amplitudes because the amplitude statistics is as important for the RPA closure as the phase statistics. Thirdly, ZS presented an argument that the nonlinear frequency correction removes the need for the initial phase randomness, whereas we only state the preservation of the initial phase randomness. However, the ZS criterion for phase randomization was obtained from a rather non-rigorous (although highly intuitive) physical argument whereas our results follow from a systematic asymptotic expansion outlined in this Letter and the details of which will be published in a more extended paper [20].
The validation of the RPA properties gives this technique the status of a well-justified 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.