On a partial data inverse problem for the semi-linear wave equation

TL;DR

This paper proves that even minimal boundary measurements uniquely determine the nonlinear term in a semi-linear wave equation on a Lorentzian manifold, using advanced linearization and Gaussian beam techniques.

Contribution

It demonstrates unique determination of the nonlinearity from arbitrarily small measurement sets without geometric restrictions, advancing inverse problem theory.

Findings

Partial Dirichlet-to-Neumann map suffices for uniqueness

No geometric or size restrictions on measurement set

Method combines higher order linearization with Gaussian beams

Abstract

We show that a partial Dirichlet-to-Neumann map, where the measurement set is arbitrarily small, uniquely determines the time-dependent nonlinearity of order three or higher in a semi-linear wave equation up to natural obstructions on a Lorentzian manifold with boundary. In particular, we do not impose any geometric or size restrictions on the measurement set. The proof relies on the technique of higher order linearization combined with the construction of Gaussian beams with reflections on the boundary.