Conformal flow on \(\mathrm{S}^{3}\) and weak field integrability in \(\mathrm{AdS}_{4}\). (English) Zbl 1378.35300
Authors’ 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 (\(\mathrm{AdS}_4\)) 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 Szegő equation, which was shown by Gérard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in \(\mathrm{AdS}_4\) are integrable as well.
Reviewer: Alessandro Selvitella (Ottawa)
MSC:
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B35 | Stability in context of PDEs |
| 83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |
| 83F05 | Relativistic cosmology |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
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