Capoferri, Matteo;
Levitin, Michael;
Vassiliev, Dmitri;
(2022)
Geometric wave propagator on Riemannian manifolds.
Communications in Analysis and Geometry
, 30
(8)
pp. 1713-1777.
10.4310/cag.2022.v30.n8.a2.
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Abstract
We study the propagator of the wave equation on a closed Riemannian manifold . We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator—a scalar function on the cotangent bundle—and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise obstructions due to caustics and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.
| Type: | Article |
|---|---|
| Title: | Geometric wave propagator on Riemannian manifolds |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.4310/cag.2022.v30.n8.a2 |
| Publisher version: | https://dx.doi.org/10.4310/CAG.2022.v30.n8.a2 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10174285 |
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