  We investigate formation of Bose-Einstein condensates under non-equilibrium
conditions using numerical simulations of the three-dimensional
Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear
excitations and the forcing at high wave numbers, and study propagation of the
turbulent spectrum toward the low wave numbers. Our primary goal is to compare
the results for the evolving spectrum with the previous results obtained for
the kinetic equation of weak wave turbulence. We demonstrate existence of a
regime for which good agreement with the wave turbulence results is found in
terms of the main features of the previously discussed self-similar solution.
In particular, we find a reasonable agreement with the low-frequency and the
high-frequency power-law asymptotics of the evolving solution, including the
anomalous power-law exponent $x^* \approx 1.24$ for the three-dimensional
waveaction spectrum. We also study the regimes of very weak turbulence, when
the evolution is affected by the discreteness of the Fourier space, and the
strong turbulence regime when emerging condensate modifies the wave dynamics
and leads to formation of strongly nonlinear filamentary vortices.
