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Sibirsk. Mat. Zh., 2006, Volume 47, Number 5, Pages 993–1018 (Mi smj926)  

This article is cited in 14 scientific papers (total in 14 papers)

The two-dimensional eikonal equation

A. V. Borovskikh

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study the two-dimensional eikonal equation $\psi^2_x+\psi^2_y=1/v^2(x,y)$. We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric $ds^2=[dx^2+dy^2]/v^2(x,y)$. We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.

Keywords: eikonal equation, inhomogeneous medium, wave front, symmetry group, equivalence group, explicit solution

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English version:
Siberian Mathematical Journal, 2006, 47:5, 813–834

Bibliographic databases:

UDC: 517.958
Received: 02.04.2005
Revised: 22.04.2006

Citation: A. V. Borovskikh, “The two-dimensional eikonal equation”, Sibirsk. Mat. Zh., 47:5 (2006), 993–1018; Siberian Math. J., 47:5 (2006), 813–834

Citation in format AMSBIB
\by A.~V.~Borovskikh
\paper The two-dimensional eikonal equation
\jour Sibirsk. Mat. Zh.
\yr 2006
\vol 47
\issue 5
\pages 993--1018
\jour Siberian Math. J.
\yr 2006
\vol 47
\issue 5
\pages 813--834

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    This publication is cited in the following articles:
    1. A. V. Borovskikh, “Gruppy ekvivalentnosti uravnenii eikonala i klassy ekvivalentnykh uravnenii”, Vestn. NGU. Ser. matem., mekh., inform., 6:4 (2006), 3–42  mathnet
    2. Kutev N, Milousheva V, “On the solvability of nonlinear elliptic systems generating minimal foliated semi-symmetric hypersurfaces”, Comptes Rendus de l Academie Bulgare Des Sciences, 60:12 (2007), 1259–1264  mathscinet  zmath  isi
    3. E. D. Moskalenskii, “Finding exact solutions to the two-dimensional eikonal equation”, Num. Anal. Appl., 2:2 (2009), 165–172  mathnet  crossref
    4. E. D. Moskalensky, “On detecting a wavefront described by 2D eikonal equation, when velocity in a medium depends on one spatial variable”, Num. Anal. Appl., 3:1 (2010), 52–58  mathnet  crossref
    5. Popovych R.O., Kunzinger M., Eshraghi H., “Admissible Transformations and Normalized Classes of Nonlinear Schrodinger Equations”, Acta Applicandae Mathematicae, 109:2 (2010), 315–359  crossref  mathscinet  zmath  isi  elib  scopus
    6. E. D. Moskalensky, “Formulas for setting a location of the wavefront propagating in a medium with power dependence of velocity on a coordinate”, Num. Anal. Appl., 4:2 (2011), 136–144  mathnet  crossref
    7. Bihlo A., Cardoso-Bihlo Elsa Dos Santos, Popovych R.O., “Complete Group Classification of a Class of Nonlinear Wave Equations”, J. Math. Phys., 53:12 (2012), 123515  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. A. V. Borovskikh, “Eikonal equation for anisotropic media”, J. Math. Sci. (N. Y.), 197:2 (2014), 248–289  mathnet  crossref  elib
    9. Molitor M., “Gaussian Distributions, Jacobi Group, and Siegel-Jacobi Space”, J. Math. Phys., 55:12 (2014), 122102  crossref  mathscinet  zmath  isi  elib  scopus
    10. Fedorchuk V., Fedorchuk V., “on Classification of Symmetry Reductions For the Eikonal Equation”, Symmetry-Basel, 8:6 (2016), 51  crossref  mathscinet  zmath  isi  scopus
    11. E. D. Moskalensky, “The novel class of exact solutions of the two-dimensional eikonal equation when the velocity in a medium depends on one spatial coordinate”, Num. Anal. Appl., 11:3 (2018), 208–219  mathnet  crossref  crossref  isi  elib  elib
    12. Lempert A., Le Q.M., “Multiple Covering of a Closed Set on a Plane With Non-Euclidean Metrics”, IFAC PAPERSONLINE, 51:32 (2018), 850–854  crossref  isi  scopus
    13. A. L. Kazakov, A. A. Lempert, K. M. Le, “O zadachakh postroeniya mnogokratnykh pokrytii i upakovok v dvumernom neevklidovom prostranstve”, UBS, 81 (2019), 6–25  mathnet  crossref
    14. An. G. Marchuk, E. D. Moskalenskii, “Semeistvo reshenii dvumernogo uravneniya eikonala”, Sib. zhurn. vychisl. matem., 23:2 (2020), 155–164  mathnet  crossref
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