  I investigate stationary scaling states of Burgers' and Gross-Pitaevskii
equations (GPE).
  The path integral representation of the steady state of the stochastic
Burgers equation is used in order to investigate the scaling solutions of the
system at renormalisation group (RG) fixed points. I employ the functional RG
in order to access the non-perturbative regime. I devise an approximation that
respects Galilei invariance and is designed to resolve the frequency and
momentum dependence of low order correlation functions. I establish a set of RG
fixed point equations for inverse propagators with an arbitrary frequency and
momentum dependence. In all spatial dimensions they yield a continuum of fixed
points as well as an isolated one. These results are fully compatible with the
existing literature for $d=1$. For $d\neq1$ however results of the literature
focus almost exclusively on irrotational solutions while the solutions that my
approximation can capture contain necessarily vorticity and are closer to
Navier-Stokes turbulence.
  Non-equilibrium steady states of ultra-cold Bose gases coupled to external
baths of energy and particles such as exciton-polariton condensates are related
to Kardar-Parisi-Zhang (KPZ) dynamics. I postulate that the scaling that we
obtain in this context applies as well to far-from-equilibrium quasi-stationary
states (non-thermal fixed points) of the closed system described by the GPE. I
translate results found in the KPZ literature to their corresponding dual in
the ultra-cold Bose gas set-up. I find that this provides a new scaling
relation which can be used to analytically identify the classical Kolmogorov
-5/3 exponent and its anomalous correction. Moreover I estimate the anomalous
correction to the scaling exponent of the compressible part of the kinetic
energy spectrum of the Bose gas which is confirmed by numerical simulations of
the GPE.
