  We test the predictions of the theory of weak wave turbulence by performing
numerical simulations of the Gross-Pitaevskii equation (GPE) and the associated
wave-kinetic equation (WKE). We consider an initial state localized in Fourier
space, and we confront the solutions of the WKE obtained numerically with GPE
data for both the wave-action spectrum and the probability density functions
(PDFs) of the Fourier mode intensities. We find that the temporal evolution of
the GPE data is accurately predicted by the WKE, with no adjustable parameters,
for about two nonlinear kinetic times. Qualitative agreement between the GPE
and the WKE persists also for longer times with some quantitative deviations
that may be attributed to the combination of breakdown of the theoretical
assumptions underlying the WKE as well as numerical issues. Furthermore, we
study how the wave statistics evolves toward Gaussianity in a time scale of the
order of the kinetic time.The excellent agreement between direct numerical
simulations of the GPE and the WKE provides a new and solid ground to the
theory of weak wave turbulence.
