Advertisement

Theoretical and Mathematical Physics

, Volume 107, Issue 2, pp 568–578 | Cite as

The elliptic-in-t solutions of the nonlinear Schrödinger equation

  • A. O. Smirnov
Article

Abstract

Four various “anzatzes” of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov,Theory of Solitons. The Method of the Inverse Scattering, Plenum, New York (1984).Google Scholar
  2. 2.
    L. D. Faddeev and L. A. Takhtajan,The Hamiltonian Methods in Theory of Solitons, Berlin-Heidelberg-New York, Springer (1987).Google Scholar
  3. 3.
    F. Calogero and A. Degasperis,Solitons and Spectral Transform I, North-Holland, Amsterdam (1982).Google Scholar
  4. 4.
    M. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1985).Google Scholar
  5. 5.
    R. Dodd, J. Eilbuck, J. Gibbon, and H. Morris,Solitons and Nonlinear Wave Equations [Russian translation], Mir, Moscow (1988).Google Scholar
  6. 6.
    A. C. Newell,Solitons in Mathematics and Physics, SIAM, Philadelphia (1985).Google Scholar
  7. 7.
    B. A. Dubrovin and S. P. Novikov,Zh. Exp. Theor. Phys.,67, 2131 (1974).Google Scholar
  8. 8.
    H. Airault, H. P. McKean, and J. Moser,Commun. Pure Appl. Math.,30, 95 (1977).Google Scholar
  9. 9.
    I. M. Krichever,Funkts. Anal. Prilozh.,14, No. 4, 45 (1980).Google Scholar
  10. 10.
    E. D. Belokolos and V. Z. Enolskii,Teor. Mat. Fiz.,53, 271 (1982).Google Scholar
  11. 11.
    M. V. Babich, A. I. Babenko, and V. B. Matveev,Izv. Akad. Nauk SSSR, ser. Mat.,49, 511 (1985).Google Scholar
  12. 12.
    E. D. Belokolos, A. I. Babenko, V. B. Matveev, and V. Z. Enolskii,Usp. Mat. Nauk,41, No. 2, 3 (1986).Google Scholar
  13. 13.
    J.-L. Verdier, in:Algebraic Analysis, Vol. II, Academic Press (1988), pp. 901.Google Scholar
  14. 14.
    A. Treibich,Duke Math. J.,59, 611 (1989).Google Scholar
  15. 15.
    A. Treibich and J.-L. Verdier,Solitons elliptiques (Volume en l'honneur du anniversaire du prof. A. Grothendieck), Birkhäuser, Boston (1990).Google Scholar
  16. 16.
    E. D. Belokolos and V. Z. Enolskii,Usp. Mat. Nauk,44, No. 5, 155 (1989).Google Scholar
  17. 17.
    E. D. Belokolos and V. Z. Enolskii,Funkts. Anal. Prilozhen.,23, No. 1, 57 (1989).Google Scholar
  18. 18.
    A. O. Smirnov,Mat. Zametki,46, No. 5, 100 (1989).Google Scholar
  19. 19.
    A. O. Smirnov,Mat. Zametki,45, No. 6, 66 (1989).Google Scholar
  20. 20.
    I. A. Taimanov,Teor. Mat. Fiz.,84, 38 (1990).Google Scholar
  21. 21.
    A. O. Smirnov,Teor. Mat. Fiz.,100, 183 (1994).Google Scholar
  22. 22.
    A. O. Smirnov,Acta Appl. Math.,36, 125 (1994).Google Scholar
  23. 23.
    I. M. Krichever,Usp. Mat. Nauk,33, No. 4, 215 (1978).Google Scholar
  24. 24.
    I. M. Krichever,Usp. Mat. Nauk,36, No. 2, 79 (1981).Google Scholar
  25. 25.
    A. O. Smirnov,Teor. Mat. Fiz.,78, 11 (1989).Google Scholar
  26. 26.
    A. O. Smirnov,Mat. Sbornik,185, No. 8, 103 (1994).Google Scholar
  27. 27.
    A. I. Babenko,Funkts. Anal. Prilozhen.,18, No. 3, 74 (1984).Google Scholar
  28. 28.
    A. O. Smirnov,Mat. Sbornik,181, No. 6, 804 (1990).Google Scholar
  29. 29.
    A. O. Smirnov,Zap. Nauchn. Sem. POMI,205 (1993).Google Scholar
  30. 30.
    A. R. Its,Vestnik Leningr. Gos. Univ., Mat. Mekh. Astr.,7, No. 2, 39 (1976).Google Scholar
  31. 31.
    A. R. Its, “The exact integration in the Riemann θ-functions of the nonlinear Schrödinger equation and of the modified Korteweg-de Vries equation,” Doctoral thesis, Leningr. Gos. Univ., Leningrad (1977).Google Scholar
  32. 32.
    A. R. Its and V. P. Kotliarov,Dokl. Akad. Nauk UkrSSR, ser. A,11, 965 (1976).Google Scholar
  33. 33.
    V. B. Matveev, “Abelian functions and solitons,” Preprint No. 373, Wroclaw, Univ. of Wroclaw (1976).Google Scholar
  34. 34.
    A. R. Its,Izv. Acad. Nauk USSR, Ser. Mat.,49, 530 (1985).Google Scholar
  35. 35.
    A. Krazer,Lehrbuch der Thetafunktionen, Teubner, Leipzig (1903).Google Scholar
  36. 36.
    N. I. Akhiezer,Elements of Elliptic Function Theory [in Russian], Nauka, Moscow (1970). Translated by V. I. Serdobol'skii.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. O. Smirnov
    • 1
  1. 1.St. Petersburg State Academy of Aerospace Instrument ProductionUSSR

Personalised recommendations