On the convergence rates of discrete solutions to the Wave Kinetic Equation

TL;DR

This paper investigates the long-term behavior of solutions to the Wave Kinetic Equation, providing quantitative convergence rates and analyzing a simplified toy model to assess the bounds' optimality.

Contribution

It offers the first quantitative convergence rates for solutions of the WKE with Dirac initial data and introduces a toy model to evaluate the bounds' optimality.

Findings

Solutions with Dirac initial data converge to a single Dirac mass at long times.

Quantitative rates for the convergence speed are established.

A toy model is analyzed to test the optimality of these bounds.

Abstract

In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Vel\'azquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions.