Inverse Boundary Value Problems for Wave Equations with Quadratic Nonlinearities

We study inverse problems for the nonlinear wave equation $\square_g u + w(x,u, \nabla_g u) = 0$inaLorentzianmanifold$(M,g)$withboundary,where$\nabla_g u$denotesthegradientand$w(x,u, \xi)$issmoothandquadraticin$\xi$. 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.