Non-equilibrium steady state for a three-mode energy cascade model

TL;DR

This paper constructs non-equilibrium steady states for a simplified three-mode energy cascade model inspired by wave turbulence, providing a new approach to understanding energy transfer in nonlinear wave systems.

Contribution

It introduces a novel method using elliptic Feynman-Kac equations to construct invariant measures for a reduced NLS model, advancing the study of energy cascades.

Findings

Successfully constructed NESS for the three-mode model

Developed a new Lyapunov function via Feynman-Kac approach

Provides insights into energy transfer mechanisms in wave turbulence

Abstract

Motivated by the central phenomenon of energy cascades in wave turbulence theory, we construct non-equilibrium statistical steady states (NESS), or invariant measures, for a simplified model derived from the nonlinear Schr\"odinger (NLS) equation with external forcing and dissipation. This new perspective to studying energy cascades, distinct from traditional analyses based on kinetic equations and their cascade spectra, focuses on the underlying statistical steady state that is expected to hold when the cascade spectra of wave turbulence manifest. In the full generality of the (infinite dimensional) nonlinear Schr\"odinger equation, constructing such invariant measures is more involved than the rigorous justification of the Kolmogorov-Zakharov (KZ) spectra, which itself remains an outstanding open question despite the recent progress on mathematical wave turbulence. Since such complexity remains far beyond the current knowledge (even for much simpler chain models), we confine our analysis to a three-mode reduced system that captures the resonant dynamics of the NLS equation, offering a tractable framework for constructing the NESS. For this, we introduce a novel approach based on solving an elliptic Feynman-Kac equation to construct the needed Lyapunov function.