Inverse Boundary Value Problems for Wave Equations with Quadratic Nonlinearities

TL;DR

This paper investigates inverse boundary value problems for nonlinear wave equations with quadratic nonlinearities on Lorentzian manifolds, demonstrating the recovery of the metric and certain nonlinear terms from boundary data.

Contribution

It introduces methods to recover the conformal class of the metric and quadratic nonlinearities from boundary measurements in nonlinear wave equations.

Findings

Conformal class of the Lorentzian metric can be recovered from boundary data.

The metric itself can be recovered up to diffeomorphisms under additional conditions.

Quadratic forms in the Taylor expansion of the nonlinearity can be identified from boundary measurements.

Abstract

We study inverse problems for the nonlinear wave equation $\square_g u + w(x,u, \nabla_g u) = 0$ in a Lorentzian manifold $(M,g)$ with boundary, where $\nabla_g u$ denotes the gradient and $w(x,u, \xi)$ is smooth and quadratic in $\xi$. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric $g$ can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of $w(x,u, \xi)$ with respect to $u$ up to null forms.