Skip to main content
SpringerLink
Log in

Huygens’ Principle and Integrability

  • Conference paper
European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 169))

Abstract

The physical notion of Huygens’Principle goes back to the classical “Traité de la Lumière” by Christian Huygens, published in 1690. Various aspects of this fundamental principle in the theory of wave propagation were later discussed in the works of Kirchhof, Poisson, Beltrami and other scientists. But it was Jacques Hadamard [1], who was the first to propose in 1923 a rigorous mathematical definition of the phenomenon he called Huygens’ Principle in the narrow sense (“minor premise”). This is the meaning of the term “Huygens’ Principle” (or, in short, HP) we use in this paper.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Hadamard Lectures on Cauchy’s Problem in Linear Partial Differential Equations. (New Haven: Yale Univ. Press, 1923).

    MATH  Google Scholar 

  2. R. Courant and D. Hibert Methods of Mathematical Physics.–V.U. (New York, 1964).

    Google Scholar 

  3. P. Günther Huygens’Principle and Hyperbolic Equations. (Boston: Acad. Press, 1988).

    Google Scholar 

  4. M. Mathisson, Le problème de M. Hadamard relatif à la diffusion des ondes, Acta Math., 71 (1939), 249–282.

    Article  MathSciNet  Google Scholar 

  5. L. Asgeirsson, Some hints on Huygens’principle and Hadamard’s conjecture, Comm. Pure Appl. Math., 9 (1956), 307–326.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Hadamard, The problem of diffusion of waves, Annals of Math., 43(3) (1942), 510–523.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Stellmacher, Ein Beispiel einer Huygensschen Differentialgleichung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl, Ha, Bd. 10 (1953), 133–138.

    MathSciNet  Google Scholar 

  8. J.E. Lagnese, K.L. Stellmacher, A method of generating classes of Huygens’operators, J. Math. & Mech. (1967), V. 17, N. 5, 461–472.

    MathSciNet  MATH  Google Scholar 

  9. J.E. Lagnese, A solution of Hadamard’s problem for a restricted class of operators, Proc. Amer. Math. Soc. (1968), V. 19, 981–988.

    MathSciNet  MATH  Google Scholar 

  10. R. Schimming Korteweg-de Vries-Hierarchie und Huygenssches Prinzip, Sitz. ber. Dresdener Seminar zur Theor. Physik (1986), Nr. 26.

    Google Scholar 

  11. H. Airault, H. McKean, and J. Moser, Rational and elliptic solutions of the KdV-equation and related many body problems, Comm. Pure Appl. Math. (1977), V. 30, 95–148.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Adler, J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys.(1978), V. 61, N. 1, 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  13. Yu.Yu. Berest, A.P. Veselov, Hadamard’s problem and Coxeter groups: new examples of the huygensian equations, Funct. Anal. Appl. (1994), V. 28, N. 1, 3–15.

    Article  MathSciNet  MATH  Google Scholar 

  14. Yu.Yu. Berest, A.P. Veselov, Huygens’principle and Integrability, Russ. Math. Surveys (1994), V. 49, N. 6, 5–77.

    Article  MathSciNet  MATH  Google Scholar 

  15. A.P. Veselov, Huygens’Principle and algebraic Schrödinger operators, Amer. Math. Soc. Transi. (2) (1995), V. 170, 199–206.

    MathSciNet  Google Scholar 

  16. O.A. Chalykh, A.P. Veselov, Integrability and Huygens’Principle on symmetric spaces, Comm. Math. Phys. (1996), V. 178, 311–338.

    Article  MathSciNet  MATH  Google Scholar 

  17. A.P. Veselov, M.V. Feigin, O.A. Chalykh, New integrable deformations of quantum Calogero-Moser problem, Uspekhi Mat. Nauk (1996), V. 51, N. 3, 185–186 (Russian).

    Article  MathSciNet  Google Scholar 

  18. Yu.Yu. Berest, I.M. Lutsenko, Huygens’principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation. To appear in Comm. Math. Phys.

    Google Scholar 

  19. Yu.Yu. Berest, Solution of a Restricted Hadamard’s Problem in Minkowski Spaces. Comm. Pure Appl. Math. (1997), V. 50, N. 10, 1021–1054.

    Article  MathSciNet  Google Scholar 

  20. S.P. Novikov, Periodic problem for KdV equation.I, Funct. Anal. Appl., (1974), V. 8, N. 3, 54–63.

    Article  Google Scholar 

  21. B.A. Dubrovin, V.B. Matveev, S.P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-gap linear operators and abelian varieties, Russian Math. Surveys. (1976), V. 31, N. 1, 51–125.

    Article  MathSciNet  Google Scholar 

  22. I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys. (1977), V. 32, N. 6, 180–208.

    Article  Google Scholar 

  23. F. Calogero, Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potential, J. Math. Phys. (1971), V. 12, 419–436.

    Article  MathSciNet  Google Scholar 

  24. J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. in Math. (1975), V. 16, 1–23.

    Article  Google Scholar 

  25. S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of solitons: The inverse scattering method, New York: Consultants Bueau, 1984.

    Google Scholar 

  26. O.A. Chalykh, A.P. Veselov, Commutative rings of partial differential operators and Lie algebras, Preprint of FIM (ETH, Zürich), 1988. Comm. Math. Phys. (1990), V. 126, 597–611.

    Article  MathSciNet  MATH  Google Scholar 

  27. O.A. Chalykh, A.P. Veselov, Integrability in the theory of the Schrödinger operators and Harmonic Analysis, Comm. Math. Phys. (1993), V. 152, 29–40.

    Article  MathSciNet  MATH  Google Scholar 

  28. A.P. Veselov, K.L. Styrkas and O.A. Chalykh, Algebraic integrability for Schrödinger equation and the groups generated by reflections, Theor. Math. Phys. (1993), V. 94, N. 2, 182–197.

    Article  MathSciNet  Google Scholar 

  29. M.A. Olshanetsky, A.M. Perelomov, Quantum completely integrable systems connected with semisimple Lie algebras, Lett. Math. Phys. (1977), V. 2, 7–13.

    Article  MathSciNet  MATH  Google Scholar 

  30. M.A. Olshanetsky, A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. (1983) V. 94, 313–404.

    Google Scholar 

  31. C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. AMS. (1989), V. 311, 167–183.

    Article  MathSciNet  MATH  Google Scholar 

  32. G.J. Heckman, A remark on the Dunkl differential-difference operators, Prog. in Math. (1991), V. 101, 181–191.

    MathSciNet  Google Scholar 

  33. E. Opdam, Root systems and hypergeometric functions IV, Compositio Math. (1988), V. 67, 191–209.

    MathSciNet  MATH  Google Scholar 

  34. A.P. Veselov, Calogero quantum problem, KZ equation and Huygens’Principle, Theor.Math.Phys. (1994), V. 98, N. 3, 524–535.

    Article  MathSciNet  Google Scholar 

  35. V.M. Buchstaber, G. Felder, A.P. Veselov, Elliptic Dunkl operators, root systems and functional equations, Duke Math. J. (1994), V. 76, N. 3, 885–911.

    Article  MathSciNet  MATH  Google Scholar 

  36. J.L. Burchnall, and T.W. Chaundy, A set of differential equations which can be solved by polynomials, Proc. London Math. Soc. (1929-30), V. 30, 401–414.

    Google Scholar 

  37. J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. (1983), V. 24, 522–526.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Schimming, Laplace-like linear differential operators with logarithm-free elementary solution, Math. Nachr. (1990), V. 148, 145–174.

    Article  MathSciNet  MATH  Google Scholar 

  39. P.D. Lax, R.S. Phillips, An example of Huygens’Principle, Comm. Pure and Appl. Math. (1978), V. 31, 415–421.

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Helgason, Wave equations on homogeneous spaces, Lect. Notes in Math. Springer-Verlag Berlin (1984), V. 1077, 252–287.

    MathSciNet  Google Scholar 

  41. L.E. Solomatina, Translation representation and Huygens’principle for an invariant wave equation in a Riemannian symmetric spaces, Soviet Math. Izv. (1986), V. 30, 108–111.

    MATH  Google Scholar 

  42. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.

    MATH  Google Scholar 

  43. G. Olafsson, H. Schlichtkrull, Wave propagation on Riemannian symmetric spaces, J. Funct. Anal. (1992), V. 107, 270–278.

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Helgason, Huygens’Principle for wave equations on Symmetric spaces, J. Funct. Anal. (1992), V. 107, 279–288.

    Article  MathSciNet  MATH  Google Scholar 

  45. T. Branson, G. Olafsson, H. Schlichtkrull, Huygens’principle in Riemannian symmetric spaces, Math. Ann. (1995), V. 301, 445–462.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Branson, G. Olafsson, Helmgoltz operator and symmetric spaces duality, Preprint, June 1995.

    Google Scholar 

  47. M. Atiyah, Green’s functions for self-dual four manifolds, Math. Analysis and Appl., Part A, Adv. in Math. Suppl. Stud. (1981), V. 7A, 129–158.

    MathSciNet  Google Scholar 

  48. P. Günther, Ein Beispiel einer nichttrivialen Huygensschen Differentialgleichung mit vier unabhängigen Variablen, Arch. Rat. Mech. Anal. (1965), V. 18(2), 103–106.

    Google Scholar 

  49. R.G. McLenaghan, An explicit determination of the empty space-times on which the wave equation satisfies Huygens’Principle, Proc. Cambridge Phil. Soc, (1969), V. 65, 139–155.

    Article  MathSciNet  MATH  Google Scholar 

  50. N.H. Ibragimov, A.O. Oganesyan, The hierarchy of Huygens’equations in spaces with a non-trivial conformai group, Russian Math. Surveys (1991), V. 46, N. 3, 111–146.

    Article  MathSciNet  Google Scholar 

  51. J.J. Duistermaat, F.A. Grünbaum, Differential equations with the spectral parameter, Comm. Math. Phys. (1986), V. 103, 177–240.

    Article  MathSciNet  MATH  Google Scholar 

  52. Yu.Yu. Berest, Nonlinear gauge transformations, Huygens’Principle and Hadamard conjecture revisited, Preprint CRM-2296 (1995).

    Google Scholar 

  53. Yu.Yu. Berest, P. Winternitz, Huygens’Principle and Separation of Variables, Preprint CRM (1996). To appear in Rev. Math. Phys.

    Google Scholar 

  54. O.A. Chalykh, M.V. Feigin, A.P. Veselov, Multidimensional Baker-Akhiezer functions and Huygens ’Principle, in preparation.

    Google Scholar 

  55. O.A. Chalykh, M.V. Feigin, A.P. Veselov, New integrable generalizations of the Calogero-Moser quantum problem, J. Math. Phys. (1998), V. 39, N. 2, 1–9.

    Article  MathSciNet  Google Scholar 

  56. A.P. Veselov, Baker-Akhiezer functions and the bispectral problem in many dimensions, CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, RI (1998), V. 14, 123–129.

    MathSciNet  Google Scholar 

  57. Yu.Yu. Berest, Huygens’principle and the bispectral problem, CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, RI (1998), V. 14, 11–30.

    MathSciNet  Google Scholar 

  58. Yu.Yu. Berest, A.P. Veselov, On the singularities of the potentials of exactly solvable Schrödinger operators and Hadamard’s problem, Uspekhi Matematicheskikh Nauk (1998), V. 53, N. 1, 211-212 (Russian).

    Google Scholar 

  59. O.A. Chalikh, Darboux transformations for multidimensional Schrödinger operators, Uspekhi Matematicheskikh Nauk (1998), V. 53, N. 2 (Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

    A. P. Veselov

  2. Department of Mathematics and Mechanics, Moscow State University, 119899, Moscow, Russia

    A. P. Veselov

Authors
  1. A. P. Veselov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Hungarian Academy of Sciences, Mathematical Institute, Reáltanoda str. 13-15, H-1053, Budapest, Hungary

    A. Balog , G. O. H. Katona  & D. Sza’sz ,  & 

  2. Mathematical Institute, Technical University of Budapest, H-1521, Budapest, Hungary

    A. Recski

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Basel AG

About this paper

Cite this paper

Veselov, A.P. (1998). Huygens’ Principle and Integrability. In: Balog, A., Katona, G.O.H., Recski, A., Sza’sz, D. (eds) European Congress of Mathematics. Progress in Mathematics, vol 169. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8898-1_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8898-1_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9819-5

  • Online ISBN: 978-3-0348-8898-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation