  The MMT equation was proposed by Majda, McLaughlin and Tabak as a model to
study wave turbulence. We focus on the kinetic equation associated to this
Hamiltonian system, which is believed to give a way to predict turbulent
spectra. We clarify the formulation of the problem, and we develop the local
well-posedness theory for this equation. Our analysis uncovers a surprising
nonlinear smoothing phenomenon.
