Recovery of Zeroth Order Coefficients in Non-Linear Wave Equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a function V from the source to solution map of the semi-linear equation ( g+V)u+u³=0 on a globally hyperbolic Lorentzian manifold (M,g) . We first study the simpler model problem, where (M,g) is the Minkowski space, and prove the unique recovery of V through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.