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We study the cubic wave equation in AdS_(d+1) (and a closely related cubicwave equation on S^3) in a weakly nonlinear regime. Via time-averaging, thesesystems are accurately described by simplified infinite-dimensional quarticHamiltonian systems, whose structure is mandated by the fully resonant spectrumof linearized perturbations. The maximally rotating sector, comprising only themodes of maximal angular momentum at each frequency level, consistentlydecouples in the weakly nonlinear regime. The Hamiltonian systems obtained bythis decoupling display remarkable periodic return behaviors closely analogousto what has been demonstrated in recent literature for a few other relatedequations (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-Pitaevskiiequation for harmonically trapped Bose-Einstein condensates.

Maximally rotating waves in AdS and on spheres