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

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.