Stability and cascades for the Kolmogorov-Zakharov spectrum of wave turbulence

TL;DR

This paper investigates the stability of Kolmogorov-Zakharov spectra in wave turbulence, demonstrating the nonlinear stability of mass cascades and constructing out-of-equilibrium steady states with energy and mass transfer.

Contribution

It provides the first partial verification of the stability of these cascades in the isotropic case, advancing understanding of wave turbulence steady states.

Findings

Proves nonlinear stability of mass cascade in stationary setting.

Constructs out-of-equilibrium steady states with cascades.

Verifies the generic development of energy and mass cascades.

Abstract

The kinetic wave equation arises in wave turbulence to describe the Fourier spectrum of solutions to the cubic Schroedinger equation. The equation has two Kolmogorov-Zakharov steady states corresponding to out-of-equilibrium cascades transferring for the first solution mass from infinity to zero (small spatial scales to large scales) and for the other solution energy from zero to infinity. After conjecturing the generic development of the two cascades, we verify it partially in the isotropic case by proving the nonlinear stability of the mass cascade in the stationary setting. This constructs non-trivial out-of-equilibrium steady states with a direct energy cascade as well as an indirect mass cascade.