Complex structure of time-periodic solutions decoded in Poincaré-Lindstedt series: the cubic conformal wave equation on ${\mathbb{S}}^3$

Abstract

This work explores the rich structure of spherically symmetric time-periodic solutions of the cubic conformal wave equation on ${\mathbb{S}}^3$. We discover that the families of solutions bifurcating from the eigenmodes of the linearised equation form patterns similar to the ones observed for the cubic wave equation. Alongside the Galerkin approaches, we study them using the new method based on the Padé approximants. To do so, we provide a rigorous perturbative construction of solutions. Due to the conformal symmetry, the solutions presented in this work serve as examples of large time-periodic solutions of the conformally coupled scalar field on the anti-de Sitter background.

arXiv comment22 pages; 7 figures. Matches the published version

References in corpus (8)

• Periodic solutions of nonlinear wave equations with general nonlinearities
• Is AdS stable?
• AdS perturbations, isometries, selection rules and the Higgs oscillator
• Conformal flow on $S^3$ and weak field integrability in AdS${}_4$
• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Breathing modes, quartic nonlinearities and effective resonant systems
• On time periodic solutions to the conformal cubic wave equation on the Einstein cylinder
• Non-linear periodic waves on the Einstein cylinder