Can we hear Kolmogorov spectra?

We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence a nonlinear wave system evolves in long-time creating a slow transfer of energy from one mode to another. We derive this kinetic equation for the spectral transfer in terms of the second order moment. We show explicitly that such a non-equilibrium theory describes the approach to an equilibrium wave spectrum, and describes also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations confirming this scenario.