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RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  18 September 2020

Ali Feizmohammadi Open the ORCID record for Ali Feizmohammadi [Opens in a new window]  and
Lauri Oksanen Open the ORCID record for Lauri Oksanen [Opens in a new window]
Ali Feizmohammadi
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK (a.feizmohammadi@ucl.ac.uk; l.oksanen@ucl.ac.uk)
Lauri Oksanen
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK (a.feizmohammadi@ucl.ac.uk; l.oksanen@ucl.ac.uk)
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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.

Keywords

Inverse problemsnon-linear wave equationsGaussian beamsLorentzian manifolds

MSC classification

Primary: 35R30: Inverse problems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 2 , March 2022 , pp. 367 - 393
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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