  We undertake a systematic, direct numerical simulation (DNS) of the
two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the
turbulent evolutions of its solutions for a variety of initial conditions and a
wide range of parameters. We find that the time evolution of this system can be
classified into four regimes with qualitatively different statistical
properties. First, there are transients that depend on the initial conditions.
In the second regime, power-law scaling regions, in the energy and the
occupation-number spectra, appear and start to develop; the exponents of these
power-laws and the extents of the scaling regions change with time and depended
on the initial condition. In the third regime, the spectra drop rapidly for
modes with wave numbers $k > k_c$ and partial thermalization takes place for
modes with $k < k_c$; the self-truncation wave number $k_c(t)$ depends on the
initial conditions and it grows either as a power of $t$ or as $\log t$.
Finally, in the fourth regime, complete-thermalization is achieved and, if we
account for finite-size effects carefully, correlation functions and spectra
are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms.
