Nonlinear periodic waves on the Einstein cylinder (Q6612315)

The authors recall that the anti-de Sitter spacetime is the maximally symmetric solution to the vacuum Einstein equations with a negative cosmological constant: \(Ric(g)=-\Lambda g\), \(\Lambda <0\). They consider the conformal cubic wave equation on the Einstein cylinder written as: \( -\partial _{t}^{2}\phi (t,\omega )+\Delta _{\mathbb{S}^{3}}\phi (t,\omega )-\phi (t,\omega )=\left\vert \phi (t,\omega )\right\vert ^{2}\phi (t,\omega )\), for a scalar field \(\phi :\mathbb{R}\times \mathbb{S}^{3}\rightarrow \mathbb{C}\), with zero initial velocity, and perturbations around the static solution \(\phi _{0}=0\). They specialize the preceding problem written as: \( (\partial _{t}^{2}+L)u=f(x,u)\), \((t,x)\in \mathbb{R}\times I\), in three cases:\N\N\begin{itemize}\N\item conformal cubic wave equation in spherical symmetry: \(Lu=-\frac{1}{ \sin^{2}(x)}\partial _{x}(\sin^{2}(x)\partial _{x}u)+u\), \(I=(0,\pi )\), \( f(x,u)=-u^{3}\),\N\item conformal cubic wave equation out of spherical symmetry in Hopf coordinates \((\eta ,\xi _{1},\xi _{2})\in \lbrack 0,\frac{\pi }{2}]\times \lbrack 0,2\pi )\times \lbrack 0,2\pi )\): \(Lu=-\partial _{x}^{2}u-(\frac{\cos x }{\sin x}-\frac{\sin x}{\cos x})\partial _{x}u+(\frac{\mu _{1}^{2}}{\sin^{2}x}+ \frac{\mu _{2}^{2}}{\cos^{2}x}+1)u-\), \(I=(0,\frac{\pi }{2})\), \(f(x,u)=-u^{3}\),\N\item Yang-Mills equation in spherical symmetry: \(Lu=-\frac{1}{\sin^{4}(x)} \partial _{x}(\sin^{4}x\partial _{x}u)+u\), \(I=(0,\pi )\), \( f(x,u)=-3u^{2}-\sin^{2}xu^{3}\).\N\end{itemize}\N\NThe authors introduce a set of frequencies verifying a certain Diophantine condition for \(0<\alpha <1/3\): \(\mathcal{W}_{\alpha }=\{\omega \in \mathbb{R} :\left\vert \omega \cdot l-\omega _{j}\right\vert \geq \frac{\alpha }{l}\), \( \forall (l,j)\in \mathbb{N}^{2}\), \(l\geq 1\), \(\omega _{j}\neq l\}\). The first main result proves the existence of time-periodic solutions to all three models bifurcating from various \(1\)-modes. Let \((\gamma ,s)\in ( \mathbb{N}\cup \{0\})\times \mathbb{R}\) satisfy appropriate assumptions depending on the case under consideration, and let \(e_{\gamma }\) be the eigenfunction to the corresponding linear operator. Also, let \(0<\alpha <1/3\) and \(\mathcal{W}_{\alpha }\) be the corresponding set of frequencies. There exists a family \(\{u_{\epsilon }:\epsilon \in \mathcal{E}_{\alpha ,\gamma }\} \) of time-periodic solutions to the problem in one of the above cases, where \(\mathcal{E}_{\alpha,\gamma }\) is an uncountable set that has \(0\) as an accumulation point. In addition, each element \(u_{\epsilon }\) satisfies: \( u_{\epsilon }\) has period \(T_{\epsilon }=2\pi /\omega _{\epsilon }\) with \( \omega _{\epsilon }\in \mathcal{W}_{\alpha }\) being \(\epsilon \)-close to \(1\) ; \(u_{\epsilon }\in H^{1}([0,T_{\epsilon }];H^{s})\); \(u_{\epsilon }\) stays close to the solution to the linearized equation with initial data \( (u_{t}=0,\partial _{t}u_{t}=0)=(\epsilon \kappa _{\gamma }e_{\gamma },0)\), for all times: \(\sup_{t\in \mathbb{R}}\left\Vert u_{\epsilon }(t,\cdot )-\Phi ^{t\omega _{\epsilon }}(\epsilon \kappa _{\gamma }e_{\gamma })\right\Vert _{H^{s}}\lesssim \epsilon /2\), where \(\kappa _{\gamma }\) is a normalization constant. For the proof in the two first cases, the authors use the original version of Theorem 2.4 in the paper by \textit{D. Bambusi} and \textit{S. Paleari} [J. Nonlinear Sci. 11, No. 1, 69--87 (2001; Zbl 0994.37040)]. The second main result proves a modified abstract version of the Bambusi-Paleari theorem that the authors apply to the third case.