  We introduce a simplified model for wave turbulence theory -- the Wick NLS,
of which the main feature is the absence of all self-interactions in the
correlation expansions of its solutions. For this model, we derive several wave
kinetic equations that govern the effective statistical behavior of its
solutions in various regimes. In the homogeneous setting, where the initial
correlation is translation invariant, we obtain a wave kinetic equation similar
to the one predicted by the formal theory. In the inhomogeneous setting, we
obtain a wave kinetic equation that describes the statistical behavior of the
wavepackets of the solutions, accounting for both the transport of wavepackets
and collisions among them. Another wave kinetic equation, which seems new in
the literature, also appears in a certain scaling regime of this setting and
provides a more refined collision picture.
