Inverse Boundary Value Problems for Wave Equations with Quadratic Nonlinearities

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.

43 pages