Jacques Smulevici

Given a quasi-periodic wave operator $\psi_{tt}-\psi_{xx}+\mathcal{B}^{xx}(\omega t,x)\partial_{xx}$, where $\mathcal{B}^{xx}:\mathbb{T}^{\nu+1} \rightarrow \mathbb{R}$ is parity preserving, reversible and small enough and where $\omega$ is diophantine, we explain how to construct \emph{null coordinates} respecting the quasi-periodicity. In these coordinates, the principal symbol of the wave operator then has constant coefficients. 

 

As an application, we give a novel proof of \emph{reducibility}, a typical element for the construction of quasi-periodic solutions to non-linear pdes, obtained very recently in a work of Berti, Feola, Procesi and Terracina on the quasi-periodically forced linear Klein-Gordon. 

 

Our main motivation for this problem comes from the possible applications to the construction of small non-linear time-periodic pertubations of the Anti-de-Sitter spacetime. This is a joint work with Athanasios Chatzikaleas.