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Acta Applicandae Mathematica

, Volume 36, Issue 1–2, pp 7–25 | Cite as

Elliptic solutions of nonlinear integrable equations and related topics

  • I. M. Krichever
Article

Abstract

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.

Mathematics subject classifications (1991)

35Q53 35Q51 14H52 14H55 

Key words

Elliptic soliton KP equation Baker-Akhiezer function 

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References

  1. 1.
    Batemen, G. and Erdely, A.:Higher Transcendental Functions, v. 2, McGraw-Hill, New York, 1953.Google Scholar
  2. 2.
    Dubrovin, B. A., Matveev, V. B., and Novikov, S. P.:Uspekhi Mat. Nauk 31(1) (1976), 55.Google Scholar
  3. 3.
    Zakharov, V. E., Manakov, S. V., Novikov, S. P., and Pitaevskii, L. P.: in S. P. Novikov (ed.),Soliton Theory: Inverse Scattering Method, Nauka, Moscow, 1980.Google Scholar
  4. 4.
    Its, A. B. and Matveev, V. B.:Teor. Mat. Fiz. 23(1) (1975), 51.Google Scholar
  5. 5.
    Krichever, I. M.:Soviet Math. Dokl. 17 (1976), 394.Google Scholar
  6. 6.
    Krichever, I. M.:Funktsional. Anal. i Prilozhen. 11(1) (1977), 15.Google Scholar
  7. 7.
    Krichever, I. M.:Uspekhi Mat. Nauk 32(6) (1977), 100.Google Scholar
  8. 8.
    Shiota, T.:Invent. Math. 83 (1986), 33.Google Scholar
  9. 9.
    Belokolos, E.D., Bobenko, A. I., Enolskii, V. Z., and Matveev, V. B.:Russian Math. Surveys 41(2) (1986), 1.Google Scholar
  10. 10.
    Airault, H., McKean, H. P., and Moser, J.:Comm. Pure Appl. Math. 30 (1977), 94.Google Scholar
  11. 11.
    Moser, J.:Adv. in Math. 16 (1976), 197.Google Scholar
  12. 12.
    Calogero, F.:Lett. Nuovo Cim. 13 (1975), 411.Google Scholar
  13. 13.
    Verdier, J. L.: New elliptic solitons, inAlgebraic Analysis 2, special volume dedicated to Prof. M. Sato on his 60th birthday, Academic Press, New York, 1988.Google Scholar
  14. 14.
    Krichever, I. M.:Funktsional. Anal. i Prilozhen. 12(1) (1978), 76.Google Scholar
  15. 15.
    Chudnovsky, D. V. and Chudnovsky, G. V.:Nuovo Cim. 40B (1977), 339.Google Scholar
  16. 16.
    Olshanetzky, M. A. and Perelomov, A. M.:Lett Nuovo Cim. 17 (1976), 97.Google Scholar
  17. 17.
    Krichever, I. M.:Funktsional. Anal, i Prilozhen. 14 (1980), 45.Google Scholar
  18. 18.
    Chudnovsky, D. V.:J. Math. Phys. 20(12) (1979), 2416.Google Scholar
  19. 19.
    Treibich, A.:Duke Math. J. 59(3) (1989), 611.Google Scholar
  20. 20.
    Alvarez, O., Killingback, T., Mangano, M., and Windy, P.:Nuclear Phys. B. 1A (1987), 189.Google Scholar
  21. 21.
    Schellekens, A. and Warner, N.:Phys. Lett. 117B (1986), 317.Google Scholar
  22. 22.
    Schellekens, A. and Warner, N.:Phys. Lett. 181B (1986), 339.Google Scholar
  23. 23.
    Witten, E.:Comm. Math. Phys. 109 (1987), 525.Google Scholar
  24. 24.
    Taubes, C. H.:Comm. Math. Phys. 122 (1989), 455.Google Scholar
  25. 25.
    Atiyah, M. and Hirzebruch, F.:Spin-Manifolds and Group Action, Essays on Topology and Related Topics 197, Springer, Heidelberg, p. 18.Google Scholar
  26. 26.
    Landveber, P. S. (ed):Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Math. 1326, Springer, Heidelberg, 1988.Google Scholar
  27. 27.
    Quillen, D.:Bull. Amer. Math. Soc. 75(6) (1969), 1293.Google Scholar
  28. 28.
    Novikov, S. P.:Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1245.Google Scholar
  29. 29.
    Ochanin, S.:Topology 26 (1987), 143.Google Scholar
  30. 30.
    Krichever, I. M.:Mat. Zametki 47(2) (1990), 34.Google Scholar
  31. 31.
    Hirzebruch, F.: Elliptic genera of levelN for complex manifolds, Preprint MPI 89-24.Google Scholar
  32. 32.
    Hirzenbruch, F.:Topological Methods in Algebraic Topology, Springer, Heidelberg, 1986.Google Scholar
  33. 33.
    Krichever, I. M.:Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 1289.Google Scholar
  34. 34.
    Krichever, I. M.:Izv. Akad. Nauk SSSR, Ser. Mat. 40 (1976), 828.Google Scholar
  35. 35.
    Dold, A.:Relations Between Ordinary and Extraordinary Cohomology, Colloq. Algebraic Topology, Aarhess, 1962.Google Scholar
  36. 36.
    Mischenko, A. S.:Mat. Zametki 4(4) (1968), 381.Google Scholar
  37. 37.
    Kasparov, G. G.:Izv. Akad. Nauk SSSR, Ser. Mat. 33 (1969), 735.Google Scholar
  38. 38.
    Mischenko, A. S.:Mat. Sbornik 80, (1969), 307.Google Scholar
  39. 39.
    Gusein-Zade, S. M., and Krichever, I. M.:Uspekhi Mat. Nauk 27(1) (1973), 245.Google Scholar
  40. 40.
    Dubrovin, B.A. and Novikov, S. P.:JETPh 67 (1974), 2131.Google Scholar
  41. 41.
    Bukhshtaber, V. M.:Izv. Akad. Nauk SSSR, Ser. Mat. 39(5) (1975), 1044.Google Scholar
  42. 42.
    Bukhshtaber, V. M.:Uspekhi Mat. Nauk 45(3) (1990), 185.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • I. M. Krichever
    • 1
  1. 1.Landau Institute for Theoretical PhysicsAcademy of SciencesMoscowRussia

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