# Wave-Turbulence Theory of four-wave nonlinear interactions

## TL;DR

This paper derives an analytical wave turbulence equation for four-wave interactions, validates it through numerical simulations of NLSE and vibrating plates, and explores phase randomness, amplitude relaxation, and intermittency phenomena.

## Contribution

It provides the first analytical derivation of the Sagdeev-Zaslavski wave turbulence equation for four-wave systems and confirms its predictions with numerical experiments.

## Key findings

- Phases rapidly become random, independent of amplitude dynamics.
- Amplitude distributions relax faster than spectral distributions.
- The pdf equation accurately describes different forcing scenarios, including intermittency.

## 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 Schroedinger 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.