  In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al.,
respectively, formally derived an infinite, coupled hierarchy of equations for
the spectral correlation functions of a system of weakly interacting nonlinear
dispersive waves with random phases in the standard kinetic limit. Analogously
to the relationship between the Boltzmann hierarchy and Boltzmann equation,
this spectral hierarchy admits a special class of factorized solutions, where
each factor is a solution to the wave kinetic equation (WKE). A question left
open by these works and highly relevant for the mathematical derivation of the
WKE is whether solutions of the spectral hierarchy are unique, in particular
whether factorized initial data necessarily lead to factorized solutions. In
this article, we affirmatively answer this question in the case of 4-wave
interactions by showing, for the first time, that this spectral hierarchy is
well-posed in an appropriate function space. Our proof draws on work of Chen
and Pavlovi\'{c} for the Gross-Pitaevskii hierarchy in quantum many-body theory
and of Germain et al. for the well-posedness of the WKE.
