Skip to main content
SpringerLink
Log in

Conservation Laws and Other Formulas for Families of Rays and Wavefronts and for the Eikonal Equation

  • Published:
Numerical Analysis and Applications Aims and scope Submit manuscript

Abstract

In the previous studies, the author has obtained conservation laws for the 2D eikonal equation in an inhomogeneous isotropic medium. These laws are divergent identities of the form div F = 0. The vector field F is expressed through a solution to the eikonal equation (the time field), the refractive index (the equation parameter), and their partial derivatives. Besides that, equivalent conservation laws (divergent identities) were found for families of rays and families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the conservation laws obtained for the 2D eikonal equation was found. In this paper, 3D analogs of the results obtained are presented: differential conservation laws for the 3D eikonal equation and conservation laws (divergent identities of the form div F = 0) for families of rays and families of wavefronts, the vector field F expressed through classical geometric characteristics of the ray curves: their Frenet basis (the unit tangent vector, principal normal, and binormal), the first curvature, and the second curvature, or through the classical geometric characteristics of the wavefront surfaces: their normal, principal directions, principal curvatures, Gaussian curvature, and mean curvature. All the results have been obtained on the basis of the general vector and geometric formulas (differential conservation laws and some formulas) obtained for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Megrabov, A.G., A Group Approach to Inverse Problems for Differential Equations, Sov. Phys. Dokl., 1989, vol. 29, no. 3, pp. 193–196.

    MathSciNet  MATH  Google Scholar 

  2. Megrabov, A.G., On One Differential Identity, Dokl. Math., 2004, vol. 69, no. 2, pp. 282–285.

    MathSciNet  Google Scholar 

  3. Megrabov, A.G., Differential Identities Relating the Laplacian, the Modulus of Gradient, and the Gradient Directional Angle of a Scalar Function, Dokl. Math., 2009, vol. 79, no. 1, pp. 136–140.

    Article  MathSciNet  Google Scholar 

  4. Megrabov, A.G., Differential Identities Relating the Modulus and Direction of a Vector Field and Euler’s Hydrodynamic Equations, Dokl. Math., 2010, vol. 82, no. 1, pp. 625–629.

    Article  MathSciNet  Google Scholar 

  5. Megrabov, A.G., Some Differential Identities and Their Applications to the Eikonal Equation, Dokl. Math., 2010, vol. 82, no. 1, pp. 638–642.

    Article  MathSciNet  Google Scholar 

  6. Megrabov, A.G., Divergence Formulas (Conservation Laws) in the Differential Geometry of Plane Curves and Their Applications, Dokl. Math., 2011, vol. 84, no. 3, pp. 857–861.

    Article  MathSciNet  Google Scholar 

  7. Megrabov, A.G., Conservation Laws in Differential Geometry of Plane Curves and for Eikonal Equation and Inverse Problems, J. Inv. Ill-Pos. Probl., 2013, vol. 21, no. 5, pp. 601–628.

    MathSciNet  MATH  Google Scholar 

  8. Megrabov, A.G., On Some Formulas for Families of Curves and Surfaces and Aminov’s Divergent Representations, Lobachevskii J. Math., 2018, vol. 39, no. 1, pp. 114–120.

    Article  MathSciNet  Google Scholar 

  9. Megrabov, A.G., Relationships between the Characteristics of Mutually Orthogonal Families of Curves and Surfaces, Bull. Novosibirsk Comp. Center. Ser. Math. Model. Geophys., 2016, no. 19, pp. 43–50.

  10. Megrabov, A.G., On the Conservation Laws fora Family of Surfaces, Bull. Novosibirsk Comp. Center. Ser. Math. Model. Geophys., 2016, no. 19, pp. 51–58.

  11. Ovsyannikov, L.V., Gruppovoi analiz differnetsial’nykh uravnenii (Group Analysis of Differential Equations), Moscow: Nauka, 1978.

    Google Scholar 

  12. Grebenev, V.N. and Medvedev, S.B., Hamiltonian Structure and Conservation Laws of Two-Dimensional Linear Elasticity Theory, Z. Angew. Math. Mech., 2016; DOI: https://doi.org/10.1002/zamm201500090.

  13. Chirkunov, Yu.A. and Medvedev, S.B., Conservation Laws for Plane Steady Potential Barotropic Flow, European J. Appl. Math., 2013, vol. 24, no. 6, pp. 789–801.

    Article  MathSciNet  Google Scholar 

  14. Kiryakov, P.P., Senashov, S.I., and Yakhno, A.N., Prilozhenie simmetrii i zakonov sokhraneniya k resheniyu differentsial’nykh uravnenii (Application of Symmetries and Conservation Laws to Solving Differential Equations), Novosibirsk: SO RAN, 2001.

    Google Scholar 

  15. Godunov, S.K. and Romenskii, E.I., Elementy mekhaniki sploshnykh sred i zakony sokhraneniya (Elements of Continuum Mechanics and Conservation Laws), Novosibirsk: Nauch. Kniga, 1998.

    Google Scholar 

  16. Dorodnitsyn, V.A., Finite-Difference Analogue of Noether’s Theorem, DAN, 1993, vol. 328, no. 6, pp. 678–682.

    MathSciNet  Google Scholar 

  17. Konovalov, A.N., Discrete Models in the Dynamic Problem of the Linear Theory of Elasticity and Conservation Laws, Diff. Ur., 2012, vol. 48, no. 7, pp. 990–996.

    Google Scholar 

  18. Bjushgens, S.S., Differentsial naya geometriya (Differential Geometry), Moscow: LKI, 2008.

    Google Scholar 

  19. Novikov, S.P. and Taimanov, I.A., Sovremennye geometricheskie struktury i polya (Contemporary Geometric Structures and Fields), Moscow: MTSNMO, 2005.

    Google Scholar 

  20. Kochin, N.E., Vektornoe ischislenie i nachala tenzornogo ischisleniya (Vector Calculus and Fundamentals of Tensor Calculus), Moscow: GONTI, 1938.

    Google Scholar 

  21. Rozhdestvenskii, B.L. and Yanenko, N.N., Sistemy kvazilineinykh uravnenii (Systems of Quasilinear Equations), Moscow: Nauka, 1978.

    Google Scholar 

  22. Aminov, Yu.A., Geometriya vektornogo polya (Vector Field Geometry), Moscow: Nauka, 1990.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090, Russia

    A. G. Megrabov

  2. Novosibirsk State Technical University, pr. K. Marksa 20, Novosibirsk, 630073, Russia

    A. G. Megrabov

Authors
  1. A. G. Megrabov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Megrabov.

Additional information

Dedicated to the blessed memory of my teacher Academician Anatolii Semenovich Alekseev

Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 4, pp. 475–488.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Megrabov, A.G. Conservation Laws and Other Formulas for Families of Rays and Wavefronts and for the Eikonal Equation. Numer. Analys. Appl. 12, 395–406 (2019). https://doi.org/10.1134/S1995423919040074

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423919040074

Keywords

Search

Navigation