  Within the spirit of fluid turbulence, we consider the one-dimensional
Majda-McLaughlin-Tabak (MMT) model that describes the interactions of nonlinear
dispersive waves. We perform a detailed numerical study of the direct energy
cascade in the defocusing regime. In particular, we consider a configuration
with large-scale forcing and small scale dissipation, and we introduce three
non- dimensional parameters: the ratio between nonlinearity and dispersion,
{\epsilon}, and the analogues of the Reynolds number, Re, i.e. the ratio
between the nonlinear and dissipative time-scales, both at large and small
scales. Our numerical experiments show that (i) in the limit of small
{\epsilon} the spectral slope observed in the statistical steady regime
corresponds to the one predicted by the Weak Wave Turbulence (WWT) theory. (ii)
As the nonlinearity is increased, the WWT theory breaks down and deviations
from its predictions are observed. (iii) It is shown that such departures from
the WWT theoretical predictions are accompanied by the phenomenon of
intermittency, typical of three dimensional fluid turbulence. We calculate the
structure-function as well as the probability density function of the wave
field at each scale and show that the degree of intermittency depends on
{\epsilon}.
