Conformal flow on $S^3$ and weak field integrability in AdS$_4$

TL;DR

This paper studies a conformally invariant wave equation on a sphere related to AdS spacetime, constructing an effective system revealing integrability features and stationary states in weak field regimes.

Contribution

It introduces the conformal flow as an effective dynamical system for the wave equation, uncovering invariant subspaces and stationary solutions, suggesting possible integrability.

Findings

Existence of low-dimensional invariant subspaces

Presence of stationary states with no energy transfer

Solutions with exactly periodic energy flows

Abstract

We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS$_4$ are integrable as well.