Stability and Cascades for the Kolmogorov–Zakharov Spectrum of Wave Turbulence

We consider the kinetic wave equation arising in wave turbulence to describe the Fourier spectrum of solutions to the cubic Schrödinger equation. This equation has two Kolmogorov–Zakharov steady states corresponding to out-of-equilibrium cascades transferring, for the first solution mass from ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document} to 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document} (small spatial scales to large scales), and for the second solution energy from 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document} to ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}. 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.