  Scaling in the dynamical properties of complex many-body systems has been of
strong interest since turbulence phenomena became the subject of systematic
mathematical studies. In this article, dynamical critical phenomena far from
equilibrium are investigated with functional renormalisation group equations.
The focus is set on scaling solutions of the stochastic driven-dissipative
Burgers equation and their relation to solutions known in the literature for
Burgers and Kardar-Parisi-Zhang dynamics. We furthermore relate superfluid as
well as acoustic turbulence described by the Gross-Pitaevskii model to known
analytic and numerical results for scaling solutions. In this way, the
canonical Kolmogorov exponent 5/3 for the energy cascade in superfluid
turbulence is obtained analytically. We also get first results for anomalous
exponents of acoustic and quantum turbulence. These are consistent with
existing experimental data. Our results should be relevant for future
experiments with, e.g., exciton-polariton condensates in solid-state systems as
well as with ultra-cold atomic gases.
