On the use of normal forms in the propagation of random waves

TL;DR

This paper investigates how correlations between Fourier coefficients of solutions to the Kadomtsev-Petviashvili II equation evolve over time, using normal form techniques to establish bounds on their growth for specific time scales.

Contribution

It introduces a novel application of normal form analysis to study the propagation of randomness and correlations in solutions of the KP-II equation.

Findings

Correlations remain small up to times of order ε^{-5/3} or ε^{-2}.

Normal form structure effectively controls correlation growth.

Provides bounds on the evolution of random initial data correlations.

Abstract

We consider the evolution of the correlations between the Fourier coeficients of a solution of the Kamdostev-Petviavshvili II equation when these coefficients are initially independent random variables. We use the structure of normal forms of the equation to prove that those correlations remain small until times of order $\varepsilon^{-5/3}$ or $\varepsilon^{-2}$ depending on the quantity considered.