Advertisement

Theoretical and Mathematical Physics

, Volume 105, Issue 2, pp 1369–1386 | Cite as

Factorization and Poisson correspondences

  • A. P. Fordy
  • A. B. Shabat
  • A. P. Veselov
Article

Abstract

The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general factorization problem. Generalizations related to energy-dependent Schrödinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows.

Keywords

Stationary Flow General Factorization Darboux Transformation Factorization Problem General Factorization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. P. Veselov,Russ. Math. Surv.,46, No. 5, 1–51 (1991).Google Scholar
  2. 2.
    J. Moser and A. P. Veselov,Commun. Math. Phys.,139, 217–243 (1991).Google Scholar
  3. 3.
    G. Darboux,C.R. Acad. Sci. Paris, 1456 (1882).Google Scholar
  4. 4.
    E. Schrödinger,Proc. R. Irish. Acad.,46A, 9–16 (1940).Google Scholar
  5. 5.
    A. P. Fordy and J. Gibbons,J. Math. Phys.,21, 2508–2510 (1980).Google Scholar
  6. 6.
    B. A. Kupershmidt and G. Wilson,Invent. Math.,62, 403–436 (1981).Google Scholar
  7. 7.
    H. Flaschka and D. W. McLaughlin, in:Bäcklund Transformations (R. M. Miura, ed.), Springer, Berlin (1976), pp. 253–295.Google Scholar
  8. 8.
    M. A. Semenov-Tian-Shanksy,RIMS,21, 1237–1260 (1985).Google Scholar
  9. 9.
    A. N. Leznov, A. B. Shabat, and R. I. Yamilov,Phys. Lett. A,174, 397–402 (1993).Google Scholar
  10. 10.
    A. P. Veselov and A. B. Shabat,Funct. Anal. Appl.,27, 10–30 (1993).Google Scholar
  11. 11.
    M. Antonowicz and A. P. Fordy,Commun. Math. Phys.,124, 465–486 (1989).Google Scholar
  12. 12.
    V. G. Drinfel'd and V. V. Sokolov,J. Sov. Math.,30, 1975–2036 (1985).Google Scholar
  13. 13.
    S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov,Theory of Solitons, Plenum, New York (1984).Google Scholar
  14. 14.
    M. Antonowicz and A. P. Fordy, in:Soliton Theory: A Survey of Results (A. P. Fordy, ed.), MUP, Manchester (1990), pp. 273–312.Google Scholar
  15. 15.
    P. Deift,Duke Math. J.,45, 267–310 (1978).Google Scholar
  16. 16.
    M. Antonowicz and A. P. Fordy,Rep. Math. Phys.,32, 223–233 (1993).Google Scholar
  17. 17.
    F. Ehlers and H. Knoerrer,Commun. Math. Helv.,57, No. 1, 1–10 (1982).Google Scholar
  18. 18.
    O. I. Bogoyavlenskii and S. P. Novikov,Funct. Anal. Appl.,10, 8–11 (1976).Google Scholar
  19. 19.
    I. M. Gelfand and L. A. Dikii,Russ. Math. Surv.,30, 73–113 (1975).Google Scholar
  20. 20.
    O. I. Mokhov,Izv. Akad. Nauk SSSR, Ser. Mat.,31, 657–664 (1988).Google Scholar
  21. 21.
    V. I. Arnol'd,Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin (1978).Google Scholar
  22. 22.
    V. Adler,Funct. Anal. Appl.,27, 141–143 (1993).Google Scholar
  23. 23.
    G. R. W. Quispel, J. A. G. Roberts, and C. J. Thompson,Phys. Lett. A,126, 419–421 (1988).Google Scholar
  24. 24.
    M. Antonowicz, A. P. Fordy, and S. Wojciechowski,Phys. Letts. A,124, 143–150 (1987).Google Scholar
  25. 25.
    A. P. Fordy and J. Gibbons,J. Math. Phys.,22, 1170–1175 (1981).Google Scholar
  26. 26.
    A. V. Mikhailov,JETP Lett.,30, 414–418 (1979).Google Scholar
  27. 27.
    A. P. Fordy and J. Gibbons,Commun. Math. Phys.,77, 21–30 (1980).Google Scholar
  28. 28.
    A. V. Mikhailov, M. A. Olshanetsky, and A. M. Perelomov,Commun. Math. Phys.,79, 473–488 (1981).Google Scholar
  29. 29.
    A. P. Fordy and J. Gibbons,Proc. R. Irish. Acad.,83A, 33–45 (1983).Google Scholar
  30. 30.
    J. E. Humphreys,Introduction to Lie Algebras and Representation Theory, Springer, Berlin (1972).Google Scholar
  31. 31.
    A. V. Mikhailov,Commun. Math. Phys.,79, 473–488 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. P. Fordy
    • 1
  • A. B. Shabat
    • 2
  • A. P. Veselov
    • 3
  1. 1.Department of Applied Mathematical Studies and Centre for Nonlinear StudiesUniversity of LeedsLeedsUK
  2. 2.Landau Institute for Theoretical PhysicsMoscowRussia
  3. 3.Department of Mathematics and MechanicsMoscow State UniversityMoscowRussia

Personalised recommendations