Inverse problems for nonlinear hyperbolic equations

Uhlmann, Günther; Zhai, Jian

doi:10.3934/dcds.2020380

Cited by 17 publications

(5 citation statements)

References 28 publications

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“…This is a central question in the study of inverse problems for wave equations, which is a subject that has attracted considerable attention in recent years (see e.g. [2,3,8,12,13,14,17,19,23,24] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map

Enciso¹,

Wrochna²,

Uhlmann³

2023

Preprint

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We prove that the Dirichlet-to-Neumann map of the linear wave equation determines the topological, differentiable and conformal structure of the underlying Lorentzian manifold, under mild technical assumptions. With more stringent geometric assumptions, the full Lorentzian structure of the manifold can be recovered as well. The key idea of the proof is to show that the singular support of the Schwartz kernel of the Dirichlet-to-Neumann map of a manifold completely determines the so-called boundary light observation set of the manifold together with its natural causal structure.

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Section: Introductionmentioning

confidence: 99%

“…In the case of (genuinely) semilinear wave equations, it has been shown in various settings that, up to natural obstructions, one can indeed recover the metric from the Dirichlet-to-Neumann map [10,12,13,19]. However, the corresponding problems for linear equations are still largely open.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map

Enciso¹,

Wrochna²,

Uhlmann³

2023

Preprint

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“…For hyperbolic equations, the recovery of time-dependent coefficients is possible for some nonlinear equations, whereas the corresponding problems for linear equations are still largely open. See [46] for an overview of the recent progress on inverse problems for nonlinear hyperbolic equations.…”

Section: Introductionmentioning

confidence: 99%

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Hintz¹,

Uhlmann²,

Zhai³

2021

Preprint

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We consider the semilinear wave equation g u + au 4 = 0, a = 0, on a Lorentzian manifold (M, g) with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric g and the coefficient a up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.We only consider boundary sources f with supp f ⊂ (0, T ) × ∂N and introduce the Dirichlet-to-Neumann (DN) map Λ g,a (measured in (0, T ) × ∂N ) defined aswhere u is the solution of (1.2). The well-posedness of the initial boundary value problem (1.2) with small Dirichlet data f ∈ C m , m ≥ 6, can be established as in [23]. Thus Λ g,a f is well-defined for such f . We will study the inverse problem or recovering the Lorentzian metric g and the nonlinear coefficient a from Λ g,a . We choose to consider the semilinear equation with a quartic nonlinear term because it requires the least amount of technicalities in the analysis of nonlinear interactions

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“…It was used to solve inverse boundary value problems by determining the Lorentzian manifold up to a conformal class in [15] and [7] by measuring the Dirichlet-to-Neumann or Neumann to Dirichlet map. For comprehensive reviews of a very active field see [12] and [14].…”

Section: Introductionmentioning

confidence: 99%

Breaking the conformal freedom of spacetime with supernova neutrino imaging

Ilmavirta¹,

Uhlmann²

2021

Preprint

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It is known that a geometric measurement of the light cones of supernovae determines the conformal class of the visible part of the spacetime. The conformal factor is physically meaningful but cannot be determined geometrically by anything with zero mass, such as the photon. We show that measuring the neutrino cones in addition to light cones completely removes this gauge freedom. We describe the physical model in great detail, including why ultrarelativistic neutrinos are the only option.

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Inverse problems for nonlinear hyperbolic equations

Cited by 17 publications

References 28 publications

Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map

Reconstruction of a Lorentzian manifold from its Dirichlet-to-Neumann map

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Breaking the conformal freedom of spacetime with supernova neutrino imaging

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