Wave-turbulence theory of four-wave nonlinear interactions

Abstract

The Sagdeev-Zaslavski (SZ) equation for wave turbulence is analytically derived, both in terms of generating function and of multi-point pdf, for weakly interacting waves with initial random phases. When also initial amplitudes are random, the one-point pdf equation is derived. Such analytical calculations remarkably agree with results obtained in totally different fashions. Numerical investigations of the two-dimensional nonlinear Schrödinger equation (NLSE) and of a vibrating plate prove that: (i) generic Hamiltonian 4-wave systems rapidly attain a random distribution of phases independently of the slower dynamics of the amplitudes, vindicating the hypothesis of initially random phases; (ii) relaxation of the Fourier amplitudes to the predicted stationary distribution (exponential) happens on a faster timescale than relaxation of the spectrum (Rayleigh-Jeans distribution); (iii) the pdf equation correctly describes dynamics under different forcings: the NLSE has an exponential pdf corresponding to a quasi-gaussian solution, like the vibrating plates, that also show some intermittency at very strong forcings.Introduction Dispersive waves are ubiquitous in nature, and their nonlinear interactions make them intriguing and challenging [1,2]. Wave Turbulence is the theory that describes the statistical properties of large numbers of incoherent interacting waves, with tools such as the wave kinetic equation analytically derived in the late sixties. This equation describes the evolution of the wave spectrum in time, when homogeneity and weak nonlinearity are assumed [3][4][5]. It has been applied to numerous phenomena, including ocean waves [6][7][8], capillary waves [9, 10] Alfvén waves [11], optical waves [12] and solid oscillations [13][14][15][16][17][18]. It is the analogue of the Boltzmann equation for classical particles and it allows the RayleighJeans equilibrium state as well as non-equilibrium solutions, in terms of Kolmogorov-Zakharov (KZ) spectra [19].To characterise the invariant measure of the dynamics, that is to find the complete statistical description concerning all quantities of interest, an important step has been taken by Sagdeev and Zaslavski [20], who obtained the Brout-Prigogine equation for the probability density function (pdf) of wave turbulence [21]. More recently, this statistical framework has been nicely revisited using the diagrammatic technique [4] and performing analytical calculations, in the 3-wave case [22][23][24]. Interestingly, many experimental and theoretical results have shown that deviations from wave-turbulence predictions can be found for rare events, e.g. intermittency [8,[25][26][27][28][29]. This seems to be the case when a more general theoretical framework [30][31][32][33][34] is required, because the nonlinearities are not small [35,36].In this communication, the complete wave-turbulence theory is developed for a fully general 4-wave system, whose hamiltonian is expressed by the following canoni-

4-wave dynamics in kinetic wave turbulence

“…The relation between the Peierls and our equations is thus discussed, showing the limit in which they coincide. Our framework also sheds some light on the issue of WT intermittency, as demonstrated by a companion paper [34], in which the equations obtained here are confirmed by numerical simulations of two 4-wave resonant Hamiltonian systems.This work is organized as follows. First, we describe our model and notation, which are consistent with previous works [1,33].…”

“…In absence of forcing and damping, P tends to the Rayleigh form (145) for any typical initial condition. This was tested numerically in [34].…”

Wave condensation with weak disorder versus beam self-cleaning in multimode fibers

“…We remark in this respect that the key assumption underlying the wave turbulence approach is the existence of a random phase among the modes rather than a genuine Gaussian statistics, as recently discussed in detail in Refs. [7,86,87]. We emphasize that in the present work the random phase of the modes is induced by the structural disorder of the medium that dominates nonlinear effects (L d L nl ).…”

“…Also note that we have deliberately chosen a small value of the condensate fraction n 0 /N 0.2 so as to avoid large deviations from Gaussianity for the fundamental mode -though the the- ory has been validated even for large condensate fractions in [41]. We recall here the recent works showing that a key assumption of the wave turbulence approach is the existence of a random phase among the modes rather than a genuine Gaussian statistics [86,87].…”

“…This rapid condensation can be explained by the perturbation on the dispersion relation due to the truncation of the parabolic potential, which modifies the regularization of wave resonances by removing the mode degeneracies, L pert kin ∼ b p L 2 nl /S 2 lmnp , see Appendix B. However, at variance with experiments of beam cleaning, this estimation of L pert kin assumes that the initial condition exhibits random phases among the modes, a feature of fundamental importance for the applicability of the wave turbulence theory [7,86,87]. In addition, this estimation assumes that the perturbation b p is maintained fixed, while in a real experiment such a perturbation may not be controlled all over the fiber length due to inherent imperfections and external perturbations, and the corresponding fluctuations contribute to the disorder term in the NLS equation, as discussed through the kinetic Eq.…”