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Mat. Sb., 2004, Volume 195, Number 4, Pages 23–64 (Mi msb813)  

This article is cited in 11 scientific papers (total in 11 papers)

Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium

A. V. Borovskikh

Voronezh State University

Abstract: The equation $(\nabla\psi)^2=1/v^2(x,y,z)$, known as the eikonal equation, is studied. This is the characteristic equation for the wave equations in an inhomogeneous medium, which plays a central role in the description of the geometry of the rays and the wave fronts. A full geometric classification of the family of eikonal equations is carried out (an equation is determined by the function $v(x,y,z)$, which has the meaning of the propagation velocity of a perturbation in the medium). In the cases of equations with linear or quadratic velocity function $v(x,y,z)$, explicit solutions – point source eikonals – are presented and the geometry of the rays is completely described.


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English version:
Sbornik: Mathematics, 2004, 195:4, 479–520

Bibliographic databases:

UDC: 517.958
MSC: Primary 35F20, 35Q60, 35A30, 58J70; Secondary 78A05, 74Jxx
Received: 06.11.2002

Citation: A. V. Borovskikh, “Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium”, Mat. Sb., 195:4 (2004), 23–64; Sb. Math., 195:4 (2004), 479–520

Citation in format AMSBIB
\by A.~V.~Borovskikh
\paper Group classification of the eikonal equation for
a~3-di\-men\-sional inhomogeneous medium
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 4
\pages 23--64
\jour Sb. Math.
\yr 2004
\vol 195
\issue 4
\pages 479--520

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    This publication is cited in the following articles:
    1. Borovskikh A.V., “The moving source phantom in geometric optics of inhomogeneous media”, Differ. Equ., 40:7 (2004), 927–933  mathnet  crossref  mathscinet  zmath  isi  elib
    2. Arrigo D.J., “Nonclassical contact symmetries and Charpit's method of compatibility”, J. Nonlinear Math. Phys., 12:3 (2005), 321–329  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. A. V. Borovskikh, “The two-dimensional eikonal equation”, Siberian Math. J., 47:5 (2006), 813–834  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. A. V. Borovskikh, “Gruppy ekvivalentnosti uravnenii eikonala i klassy ekvivalentnykh uravnenii”, Vestn. NGU. Ser. matem., mekh., inform., 6:4 (2006), 3–42  mathnet
    5. Vaneeva O.O., Johnpillai A.G., Popovych R.O., Sophocleous C., “Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities”, J. Math. Anal. Appl., 330:2 (2007), 1363–1386  crossref  mathscinet  zmath  isi  elib
    6. 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
    7. Alexander Bihlo, Elsa Dos Santos Cardoso-Bihlo, Roman O. Popovych, “Complete group classification of a class of nonlinear wave equations”, J. Math. Phys, 53:12 (2012), 123515  crossref  mathscinet  zmath  isi
    8. Vaneeva O.O., Popovych R.O., Sophocleous C., “Extended Group Analysis of Variable Coefficient Reaction-Diffusion Equations with Exponential Nonlinearities”, J. Math. Anal. Appl., 396:1 (2012), 225–242  crossref  mathscinet  zmath  isi
    9. A. V. Borovskikh, “Eikonal equation for anisotropic media”, J. Math. Sci. (N. Y.), 197:2 (2014), 248–289  mathnet  crossref  elib
    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. M. V. Neschadim, “Obobschennye funktsionalno-invariantnye resheniya volnovogo uravneniya v razmernosti 2”, Sib. zhurn. chist. i prikl. matem., 17:3 (2017), 58–66  mathnet  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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