  Using the 1D Majda-McLaughlin-Tabak model as an example, we develop numerical
experiments to study the validity of the Wave Kinetic Equation (WKE) at the
kinetic limit (i.e., small nonlinearity and large domain). We show that the
dynamics converges to the WKE prediction, in terms of the closure model and
energy flux, when the kinetic limit is approached. When the kinetic limit is
combined with a process of widening the inertial range, the theoretical
Kolmogorov constant can be recovered numerically to a very high precision.
