An energy cascade finite volume scheme for a mixed 3- and 4-wave kinetic equation arising from the theory of finite-temperature trapped Bose gases

TL;DR

This paper introduces a finite-volume numerical scheme for a complex wave kinetic equation involving both 3-wave and 4-wave interactions, relevant to finite-temperature Bose gases, and demonstrates its effectiveness in capturing energy cascade phenomena.

Contribution

It develops the first finite-volume scheme for a mixed 3- and 4-wave kinetic equation, extending previous methods to more complex collision operators in Bose gas models.

Findings

Successfully captures energy cascade behavior in simulations

Establishes theoretical properties of the numerical scheme

Extends numerical methods to more complex wave interactions

Abstract

Building on recent developments in numerical schemes designed to capture energy cascades for 3-wave kinetic equations~\cite{das2024numerical, walton2022deep, walton2023numerical, walton2024numerical}, we construct in this work a finite-volume algorithm for a significantly more complex wave kinetic equation whose collision operator incorporates both 3-wave and 4-wave interactions. This model arises in the context of finite-temperature Bose-Einstein condensation. We establish theoretical properties of the proposed scheme, and our numerical experiments demonstrate that it successfully captures the energy cascade behavior predicted by the equation.