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

TL;DR

This paper demonstrates that the Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds allows for the recovery of the metric and coefficient, using advanced wave interaction and scattering techniques.

Contribution

It introduces a method to recover the Lorentzian metric and nonlinear coefficient from boundary measurements for a semilinear wave equation.

Findings

Recovery of metric and coefficient from boundary data

Use of distorted plane waves and Gaussian beams

Analysis of wave interactions on Lorentzian manifolds

Abstract

We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, 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.