Wave turbulence and collective behavior models for wave equations with short- and long-range interactions

TL;DR

This paper explores how wave equations with short- and long-range interactions can lead to wave turbulence and collective behavior, introducing a new Vlasov-type kinetic model in the mean field limit.

Contribution

It develops a novel kinetic model that bridges wave turbulence and collective behavior for wave equations with non-local interactions.

Findings

Derivation of a Vlasov-type kinetic model for long-range interactions

Connection between wave turbulence and collective behavior models

Framework for analyzing wave equations with non-local operators

Abstract

In this work, we discuss a situation which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local differential operators (short-range interactions) by non-local ones (long-range interactions), we arrive at a new Vlasov-type kinetic model in the mean field limit under the molecular chaos assumption reminiscent of models for collective behavior in which anharmonic oscillators replace individual particles.