Maximally rotating waves in AdS and on spheres

TL;DR

This paper investigates the behavior of maximally rotating modes in cubic wave equations on AdS and spheres, revealing decoupling and periodic dynamics suggestive of integrability in a weakly nonlinear regime.

Contribution

It introduces a decoupling of maximally rotating modes in cubic wave equations on AdS and spheres, uncovering periodic behaviors and potential integrability in the weakly nonlinear limit.

Findings

Maximally rotating modes decouple in the weakly nonlinear regime.

The resulting Hamiltonian systems exhibit periodic return behaviors.

Connections are made to integrability and Bose-Einstein condensates.

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.