We study the cubic wave equation in AdS_(d+1) (and a closely related cubic wave equation on S^3) in a weakly nonlinear regime. Via time-averaging, these systems are accurately described by simplified infinite-dimensional quartic Hamiltonian systems, whose structure is mandated by the fully resonant spectrum of linearized perturbations. The maximally rotating sector, comprising only the modes of maximal angular momentum at each frequency level, consistently decouples in the weakly nonlinear

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... e. The Hamiltonian systems obtained by this decoupling display remarkable periodic return behaviors closely analogous to what has been demonstrated in recent literature for a few other related equations (the cubic Szego equation, the conformal flow, the LLL equation). This suggests a powerful underlying analytic structure, such as integrability. We comment on the connection of our considerations to the Gross-Pitaevskii equation for harmonically trapped Bose-Einstein condensates.