RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS. (18th March 2022)

• Record Type: Journal Article
• Authors: Feizmohammadi, Ali; Oksanen, Lauri
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 2(2022)
• Page Start: 367
• Page End: 393
• Publication Date: 2022-03-18
• Subjects: Inverse problems -- non-linear wave equations -- Gaussian beams -- Lorentzian manifolds; 35R30; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748020000122
• Languages: English
• ISSNs: 1474-7480

Abstract

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.