Null coordinates for quasi-periodic $(1+1)$-dimensional wave operators on the circle with applications to reducibility

Abstract

Given any wave operator with principle part $\partial_{t}^2 -\partial_{x}^2 +\mathcal{B}^{xx}(\omega t,x)\partial_{x}^2$, where $\mathcal{B}^{xx}:\mathbb{T}^{\nu+1} \rightarrow \mathbb{R}$ is a sufficiently small, quasi-periodic perturbation and $\omega \in \mathbb{R}^\nu$, we explain how to construct \emph{null coordinates} that respect the quasi-periodicity of the solutions. As it turns out, in these coordinates, the principal symbol of the wave operator above has constant coefficients. To construct these coordinates, we start by writing the wave operator in geometric form, modulo terms of order $1$, meaning as the wave operator arising from an $(1+1)$-Lorentzian metric and then define null coordinates as solutions to the Eikonal equations, so that the metric is conformally flat in these coordinates. The problem of constructing these coordinates is then eventually reduced to that of straightening a vector field with quasi-periodic coefficients. As an application, we give a novel proof of a recent reducibility result of Berti-Feola-Procesi-Terracina for the quasi-periodically forced linear Klein-Gordon equation, the novelty concerning essentially the analysis of the maximal order terms. In particular, the method we propose here does not rely on any quantitative Egorov-type result.