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This article is cited in 1 scientific paper (total in 1 paper)
Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation
A. G. Megrabovab a Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch,
Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090 Russia
b Novosibirsk State Technical University, pr. K. Marksa 20, Novosibirsk, 630073 Russia
Abstract:
In the previous studies, the author has obtained the conservation laws for the $2\mathrm{D}$ eikonal equation in an
inhomogeneous isotropic medium. These laws represent the divergent identities of the form $\mathrm{div}\, F = 0$. The
vector field $F$ is expressed in terms of the solution to the eikonal equation (the time field), the refractive
index (the equation parameter) and their partial derivatives. Also, there were found equivalent conservation
laws (divergent identities) for the families of rays and the families of wavefronts in terms of their geometric
characteristics. Thus, the geometric essence (interpretation) of the above-mentioned conservation laws for the
$2\mathrm{D}$ eikonal equation was discovered.
In this paper, the $3\mathrm{D}$ analogs to the results obtained are presented: differential conservation laws for the
$3\mathrm{D}$ eikonal equation and the conservation laws (divergent identities of the form $\mathrm{div}\, F = 0$) for the family of rays
and the family of wavefronts, the vector field $F$ is expressed in terms of classical geometric characteristics of
the ray curves: their Frenet basis (unit tangent vector, a principal normal and a binormal), the first curvature
and the second curvature, or in terms of the classical geometric characteristics of the wavefront surfaces, i. e.
their normal, principal directions, principal curvatures, the Gaussian curvature and the mean curvature.
All the results have been obtained based on the vector and geometric formulas (differential conservation
laws and some formulas) obtained for the families of arbitrary smooth curves, the families of arbitrary smooth
surfaces and arbitrary smooth vector fields.
Key words:
kinematic seismic, geometric optics, eikonal equation, family of rays, family of wavefronts, conservation laws, differential geometry, geometry of vector fields.
Received: 26.09.2018 Revised: 13.11.2018 Accepted: 25.07.2019
Citation:
A. G. Megrabov, “Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation”, Sib. Zh. Vychisl. Mat., 22:4 (2019), 483–497; Num. Anal. Appl., 12:4 (2019), 395–406
Linking options:
https://www.mathnet.ru/eng/sjvm727 https://www.mathnet.ru/eng/sjvm/v22/i4/p483
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Abstract page: | 126 | Full-text PDF : | 21 | References: | 15 | First page: | 3 |
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