Nonlinear periodic waves on the Einstein cylinder

Abstract

Motivated by the study of small amplitude nonlinear waves in the anti-de Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically symmetric Yang–Mills equations on the Einstein cylinder $ℝ\times {\mathbb{𝕊}}^3$. For the conformal cubic wave equation, we consider both spherically symmetric solutions and complex-valued aspherical solutions with an ansatz relying on the Hopf fibration of the $3$-sphere. In all three cases, the equations reduce to $1$+$1$ semilinear wave equations.

Our proof relies on a theorem of Bambusi–Paleari for which the main assumption is the existence of a seed solution, given by a nondegenerate zero of a nonlinear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the nondegeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials.

In the Yang–Mills case, the original version of the theorem of Bambusi–Paleari is not applicable because the nonlinearity of smallest degree is nonresonant. The resonant terms are then provided by the next order nonlinear terms with an extra correction due to backreaction terms of the smallest degree of nonlinearity, and we prove an analogous theorem in this setting.