GLORIA

GEOMAR Library Ocean Research Information Access

X

Ihre E-Mail wurde erfolgreich gesendet. Bitte prüfen Sie Ihren Maileingang.

Leider ist ein Fehler beim E-Mail-Versand aufgetreten. Bitte versuchen Sie es erneut.

Vorgang fortführen?

Exportieren
Filter
  • Cambridge University Press (CUP)  (2)
Materialart
Verlag/Herausgeber
  • Cambridge University Press (CUP)  (2)
Person/Organisation
Sprache
Erscheinungszeitraum
  • 1
    Online-Ressource
    Online-Ressource
    Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers
    Cambridge University Press (CUP) ; 2021
    In:  Forum of Mathematics, Pi Vol. 9 ( 2021)
    Details
    In: Forum of Mathematics, Pi, Cambridge University Press (CUP), Vol. 9 ( 2021)
    Kurzfassung: The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension $n+1\geqslant 3$ and with partial data. We consider the case when the set $\Omega _{\mathrm{in}}$ , where the sources are supported, and the set $\Omega _{\mathrm{out}}$ , where the observations are made, are separated. As model problems we study both a quasi-linear equation and 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 geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\in \Omega _{\mathrm{in}}$ and the past of the point $p_{out}\in \Omega _{\mathrm{out}}$ . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in $\Omega _{\mathrm{in}}$ and observations in $\Omega _{\mathrm{out}}$ , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.
    Materialart: Online-Ressource
    ISSN: 2050-5086
    URL: Article
    DOI: 10.1017/fmp.2021.11
    Sprache: Englisch
    Verlag: Cambridge University Press (CUP)
    Publikationsdatum: 2021
    ZDB Id: 2723153-7
    Standort Signatur Einschränkungen Verfügbarkeit
    BibTip Andere fanden auch interessant ...
    Online-Ressource
  • 2
    Online-Ressource
    Online-Ressource
    RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS
    Cambridge University Press (CUP) ; 2022
    In:  Journal of the Institute of Mathematics of Jussieu Vol. 21, No. 2 ( 2022-03), p. 367-393
    Details
    In: Journal of the Institute of Mathematics of Jussieu, Cambridge University Press (CUP), Vol. 21, No. 2 ( 2022-03), p. 367-393
    Kurzfassung: 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.
    Materialart: Online-Ressource
    ISSN: 1474-7480 , 1475-3030
    URL: Article
    DOI: 10.1017/S1474748020000122
    Sprache: Englisch
    Verlag: Cambridge University Press (CUP)
    Publikationsdatum: 2022
    ZDB Id: 2092996-1
    SSG: 17,1
    Standort Signatur Einschränkungen Verfügbarkeit
    BibTip Andere fanden auch interessant ...
    Online-Ressource
Schließen ⊗
Diese Webseite nutzt Cookies und das Analyse-Tool Matomo. Weitere Informationen finden Sie hier...