On the propagation of weakly nonlinear random dispersive waves

TL;DR

This paper investigates how weakly nonlinear dispersive waves with random initial conditions evolve over time, focusing on the decorrelation of Fourier modes and the influence of resonances and initial data.

Contribution

It provides new insights into the decorrelation behavior of Fourier modes in dispersive models with random initial data, considering resonance effects and initial data choices.

Findings

Fourier modes' decorrelation depends on resonances

Initial data choice influences decorrelation behavior

Resonance structures significantly affect wave evolution

Abstract

We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast Physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data.