Recovery of zeroth order coefficients in non-linear wave equations

Abstract

This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $({\square }_g+V)u+u^3=0$ on a globally hyperbolic Lorentzian manifold $(M,g)$. We first study the simpler model problem where the geometry is the Minkowski space and prove the uniqueness 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.

References in corpus (3)

• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Detection of Hermitian connections in wave equations with cubic non-linearity
• Determination of vacuum space-times from the Einstein-Maxwell equations

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• Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds
• An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds
• Uniqueness Result For Semi-linear Wave Equations With Sources