The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Abstract

We consider the semilinear wave equation ${\square }_{g}u+a{u}^4=0$, $a\ne 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.