Physical Review E
The nonlinear quantum kinetic equation for the interaction of sound waves is solved via analytic and numerical techniques. In the classical regime energy cascades to higher frequency (ω) according to the steady-state power law ω-3/2. In the quantum limit, the system prefers a reverse cascade of energy which follows the power law ω-6. Above a critical flux, a new type of spectrum appears which is neither self-similar nor close to equilibrium. This state of nonlinear quantum wave turbulence represents a flow of energy directly from the classical source to the quantum degrees of freedom.
Alejandro Garcia, M. Haeri, S. Putterman, and P. Roberts. "Quantam wave turbulence" Physical Review E (1993): 739-742. https://doi.org/http://dx.doi.org/10.1103/PhysRevE.47.739