RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS. (18th March 2022) - Feizmohammadi, Ali; Oksanen, Lauri
Abstract: 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 $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}}, g)$ . We first study the simpler model problem, where $({\mathcal{M}}, g)$ 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.