Geodesic
Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 483-497 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation},
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A. G. Megrabov. Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 483-497. http://geodesic.mathdoc.fr/item/SJVM_2019_22_4_a6/

[1] A. G. Megrabov, “O nekotorom gruppovom podkhode k obratnym zadacham dlya differencial'nykh uravneniy”, Dokl. AN SSSR, 275:3 (1984), 583–586 | MR | Zbl

[2] Megrabov A. G., “On one differential identity”, Dokl. Math., 69:2 (2004), 282–285 | MR | Zbl

[3] Megrabov A. G., “Differential identities relating the Laplacian, the modulus of gradient, and the gradient directional angle of a scalar function”, Dokl. Math., 79:1 (2009), 136–140 | DOI | MR | Zbl

[4] Megrabov A. G., “Differential identities relating the modulus and direction of a vector field and Euler's hydrodynamic equations”, Dokl. Math., 82:1 (2010), 625–629 | DOI | MR | Zbl

[5] Megrabov A. G., “Some differential identities and their applications to the eikonal equation”, Dokl. Math., 82:1 (2010), 638–664 | DOI | MR | Zbl

[6] A. G. Megrabov, “Divergence formulas (conservation laws) in the differential geometry of plane curves and their applications”, Dokl. Math., 84:3 (2011), 857–861 | DOI | MR | Zbl

[7] A. G. Megrabov, “Conservation laws in differential geometry of plane curves and for eikonal equation and inverse problems”, J. Inv. Ill-Posed Problems, 21:5 (2013), 601–628 | DOI | MR | Zbl

[8] A. G. Megrabov, “On some formulas for families of curves and surfaces and Aminov's divergent representations”, Lobachevskii J. Math., 39:1 (2018), 114–120 | DOI | MR | Zbl

[9] A. G. Megrabov, “Relationships between the characteristics of mutually orthogonal families of curves and surfaces”, Bull. Novosibirsk Comp. Center. Ser. Mathematical Modeling in Geophysics, 2016, no. 19, 43–50 | Zbl

[10] A. G. Megrabov, “On the conservation laws for a family of surfaces”, Bull. Novosibirsk Comp. Center. Ser. Mathematical Modeling in Geophysics, 2016, no. 19, 51–58 | Zbl

[11] L. V. Ovsyannikov, Gruppovoy analiz differencial'nykh uravneniy, Nauka, M., 1978

[12] V. N. Grebenev, S. B. Medvedev, “Hamiltonian structure and conservation laws of two-dimensional linear elasticity theory”, Z. Angew. Math. Mech., 2016 | DOI | MR

[13] Yu. A. Chirkunov, S. B. Medvedev, “Conservation laws for plane steady potential barotropic flow”, European J. Appl. Math., 24:6 (2013), 789–801 | DOI | MR | Zbl

[14] P. P. Kiryakov, S. I. Senashov, A. N. Yakhno, Prilozhenie simmetriy i zakonov sokhraneniya k resheniyu differencial'nykh uravneniy, Izd-vo SO RAN, Novosibirsk, 2001

[15] S. K. Godunov, E. I. Romenskiy, Elementy mekhaniki sploshnykh sred i zakony sokhraneniya, Nauchnaya kniga, Novosibirsk, 1998

[16] V. A. Dorodnicyn, “Konechno-raznostnyy analog teoremy Neter”, DAN, 328:6 (1993), 678–682 | Zbl

[17] A. N. Konovalov, “Diskretnye modeli v dinamicheskoy zadache lineynoy teorii uprugosti i zakony sokhraneniya”, Differencial'nye uravneniya, 48:7 (2012), 990–996 | MR | Zbl

[18] S. S. Byushgens, Differencial'naya geometriya, Izd-vo LKI, M., 2008

[19] S. P. Novikov, I. A. Taymanov, Sovremennye geometricheskie struktury i polya, Izd-vo MCNMO, M., 2005

[20] N. E. Kochin, Vektornoe ischislenie i nachala tenzornogo ischisleniya, GONTI, M.–L., 1938 | MR

[21] B. L. Rozhdestvenskiy, N. N. Yanenko, Sistemy kvazilineynykh uravneniy, Nauka, M., 1978 | MR

[22] Yu. A. Aminov, Geometriya vektornogo polya, Nauka, M., 1990