Inverse boundary value problems for wave equations with quadratic nonlinearities

Abstract

We study inverse problems for the nonlinear wave equation □gu+w(x,u,∇gu)=0 in a Lorentzian manifold (M,g) with boundary, where ∇gu denotes the gradient and w(x,u,ξ) is smooth and quadratic in ξ. 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,ξ) with respect to u up to null forms.