Journal of Nonlinear Science

, Volume 4, Issue 1, pp 1–21 | Cite as

Finite-gap solutions of the Davey-Stewartson equations

  • T. M. Malanyuk
Article

Summary

We describe the finite-gap solutions of different modifications of the Davey-Stewartson (DS) equations. The restrictions on the spectral data which give us solutions of the real forms DS1 and DS2+ of DS are the same as those in the case of KP1 and KP2 of the Kadomtsev-Petviashvily equation. But for DS2 the restrictions that we regard have no analogues in other integrable systems. We describe also the restrictions that provide regularity of those solutions for DS1 and DS2±. The finite-gap solutions include rational and soliton solutions. We give some classes of those solutions. The well-known dromions for DS1 are examples of that kind.

Key words

Davey-Stewartson equations finite-gap solution algebraic curve anti-involution theta-function soliton dromion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Davey, A., and Stewartson, V., Proc. R. Soc. London Ser. A 338 (1974).Google Scholar
  2. 2.
    Novikov, S.P. (ed.),Soliton Theory. Method of Inverse Problem, M. Nauka (1979) p. 289.Google Scholar
  3. 3.
    Zakharov, V.E., and Kuznetsov, E.A., Physica D 18 (1986).Google Scholar
  4. 4.
    Kadomtsev, B.B., and Petviashvily, B.I., Dokl. Acad. Nauk SSSR 31, no. 1 (Russian).Google Scholar
  5. 5.
    Krichever, I.M., Usp. Mat. Nauk. 33, no. 6 (1977) (Russian).Google Scholar
  6. 6.
    Segur, H. and Finkel, A., Stud. in Appl. Math. 73 (1985) pp. 183–220.MathSciNetGoogle Scholar
  7. 7.
    Krichever, I.M., Dokl. Acad. Nauk SSSR 227, no. 2 (1976) (Russian).Google Scholar
  8. 8.
    Dubrovin, B.A., and Natanzon, S.M., Izv. Acad. Nauk SSSR Ser. Math. 52, no. 2 (1988) (Russian).Google Scholar
  9. 9.
    Boiti, M., Leon, J., Martina, L., and Pempinelli, F., Phys. Lett. A 132 (1988) pp. 432–439.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fokas, A.S., and Santini, P.M., Physica D 44 (1990) pp. 99–130.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hietarinta, R., Hirota, R., Phys. Lett. A 145 (1990) p. 237.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Arkadiev, V.A., Pogrebkove, A.K., and Polivanov, M.C., Physica D 36 no. 1/2 (1989).Google Scholar
  13. 13.
    Freman, N.C., IMA. J. Appl. Math. 26 (1983) pp. 2916–2918Google Scholar
  14. 14.
    Beals, R., and Coifman, R.R., Physica D 18 (1986) pp. 242–249.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fay, I.D.,Theta Functions on Riemann Surfaces. Lect. Notes Math. 1352, Berlin-Heidelberg-New York, Springer (1973).Google Scholar
  16. 16.
    Dubrovin, B. A., Usp. Mat. Nauk. 36, no. 6 (1981) (Russian).Google Scholar
  17. 17.
    Natanzon, S.M., Dokl. Acad. Nauk SSSR 297, no. 1 (1987) (Russian).Google Scholar
  18. 18.
    Dubrovin, B.A., Krichever, I.M., Malanyuk, T.M., and Makhankov, V.G., Phys. Elem. Chast. & Atom. Jadra 19, no. 3 (1988) (Russian).Google Scholar

Copyright information

© Springer-Verlag New York Inc 1994

Authors and Affiliations

  • T. M. Malanyuk
    • 1
  1. 1.Department of Physics and Mathematics, Chair of InformaticsTernopol Pedagogical InstituteTernopolUkraine

Personalised recommendations