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

We consider the semilinear wave equation □gu+au4=0, a≠0, on a Lorentzian manifold (M, g) with timelike boundary. show that from knowledge of Dirichlet-to-Neumann map one can recover metric g and coefficient up to natural obstructions. Our proof rests analysis interaction distorted plane waves together scattering control argument, as well Gaussian beam solutions.