Inverse problems for non-linear hyperbolic equations with disjoint sources and receivers

Abstract

The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that makes it possible to study the inverse problem in all dimensions $n+1\ge 3$. We consider the case when the set ${\Omega }_{\text{in}}$, where the sources are supported, and the set ${\Omega }_{\text{out}}$, where the observations are made, are separated. As model problems we study both a quasi-linear and also a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the first half we study multiple-fold linearization of the non-linear wave equation near real parts of Gaussian beams that results in a three-wave interaction. We show that the three-wave interaction can produce a three-to-one scattering data. In the second half of the paper, we study an abstract formulation of the three-to-one scattering relation showing that it recovers the topological, differential and conformal structures of the manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\in {\Omega }_{\text{in}}$ and the past of the point $p_{out}\in {\Omega }_{\text{out}}$. The results do not require any assumptions on the conjugate or cut points.

61 pages

References in corpus (6)

• 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
• Inverse Scattering for Critical Semilinear Wave Equations
• Inverse problem for the Yang-Mills equations
• Inverse problems for real principal type operators

Cited by in corpus (2)

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