  A mathematical framework is presented to study the evolution of multi-point
cumulants in nonlinear dispersive partial differential equations with random
input data, based on the theory of weak wave turbulence (WWT). This framework
is used to explain how energy is distributed among Fourier modes in the
nonlinear Schr\"odinger equation. This is achieved by considering interactions
among four Fourier modes and studying the role of the resonant, non-resonant,
and trivial quartets in the dynamics. As an application, a power spectral
density is suggested for calculating the interference power in dense
wavelength-division multiplexed optical systems, based on the kinetic equation
of the WWT. This power spectrum, termed the Kolmogorov-Zakharov (KZ) model,
results in a better estimate of the signal spectrum in optical fiber, compared
with the so-called Gaussian noise (GN) model. The KZ model is generalized to
non-stationary inputs and multi-span optical systems.
