Title: Maximally rotating waves in AdS and on spheres,
Abstract: 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 regime. 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.