Functional Analysis and Its Applications

, Volume 38, Issue 2, pp 88–101

Heat Equations in a Nonholomic Frame

  • V. M. Buchstaber
  • D. V. Leykin
Article

Abstract

A system of heat equations in a nonholonomic frame is considered. Solutions of the system are constructed in the form of general sigma functions of Abelian tori. As a corollary, we solve the problem (of general interest) to describe the generators of the ring of differential operators annihilating the sigma functions of families of plane algebraic curves.

nonholonomic frame heat equations sigma and theta functions in several variables discriminant varieties 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Academic Publishers, Dordrecht, 1990.Google Scholar
  2. 2.
    H. F. Baker, Abelian Functions, Cambridge University Press, Cambridge, 1995.Google Scholar
  3. 3.
    H. F. Baker, An Introduction to the Theory of Multiply Periodic Functions, Cambridge University Press, Cambridge, 1907.Google Scholar
  4. 4.
    O. Bolza, “The partial differential equations for the hyperelliptic θ and σ-functions,” Amer. J. Math., 21, 107–125 (1899).Google Scholar
  5. 5.
    O. Bolza, “Remarks concerning the expansions of the hyperelliptic σ-functions,” Amer. J. Math., 22, 101–112 (1900).Google Scholar
  6. 6.
    V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Kleinian functions, hyperelliptic Jacobians and applications,” Rev. Math. Math. Phys., 10, No. 2, 3–120 (1997).Google Scholar
  7. 7.
    V. M. Bukhshtaber, D. V. Leikin, and V. Z. Enol'skii, “Rational analogues of Abelian functions,” Funkts. Anal. Prilozhen., 33, No. 2, 1–15 (1999).Google Scholar
  8. 8.
    V. M. Bukhshtaber and D. V. Leikin, “Lie algebras associated with σ-functions and versal deformations,” Usp. Mat. Nauk, 57, No. 3, 145–146 (2002).Google Scholar
  9. 9.
    V. M. Bukhshtaber and D. V. Leikin, “Graded Lie algebras that define hyperelliptic sigma functions,” Dokl. RAS, 385, No. 5, 583–586 (2002).Google Scholar
  10. 10.
    V. M. Bukhshtaber and D. V. Leikin, “Polynomial Lie algebras,” Funkts. Anal. Prilozhen., 36, No. 4, 2002, 18–34 (2002).Google Scholar
  11. 11.
    F. Frobenius und L. Stickelberger, “Ñber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten,” J. Reine Angew. Math., 92, 311–327 (1882).Google Scholar
  12. 12.
    F. Klein, “Ñber hyperelliptische Sigmafunktionen,” Gesammelte Mathematische Abhandlungen, Vol. 3, Teubner, Berlin, 1923, pp. 323–387.Google Scholar
  13. 13.
    A. Krazer, “Zur Bildung allgemeiner σ-Functionen,” Math. Ann., 33, 591–599 (1889).Google Scholar
  14. 14.
    K. Weierstrass, “Zur Theorie der elliptischen Functionen,” Mathematische Werke, Vol. 2, Teubner, Berlin, 1894, pp. 245–255.Google Scholar
  15. 15.
    E. Wiltheiss, “Partielle Differentialgleichungen der hyperelliptischen Thetafunctionen und der Perioden derselben,” Math. Ann., 31, 134–155 (1888).Google Scholar
  16. 16.
    E. Wiltheiss, “Ñber die Potenzreihen der hyperelliptischen Thetafunctionen,” Math. Ann., 31, 410–423 (1888).Google Scholar
  17. 17.
    V. M. Zakalyukin, “Rearrangements of wave fronts that depend on a certain parameter,” Funkts. Anal. Prilozhen., 10, No. 2, 69–70 (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • V. M. Buchstaber
    • 1
  • D. V. Leykin
    • 2
  1. 1.V. A. Steklov Mathematical instituteRussian Academy of SciencesRussia
  2. 2.Institute of MagnetismNational Academy of Sciences of UkraineUkraine

Personalised recommendations