Theoretical and Mathematical Physics

, Volume 129, Issue 2, pp 1448–1465 | Cite as

Lagrangian Chains and Canonical Bäcklund Transformations

  • V. E. Adler
  • V. G. Marikhin
  • A. B. Shabat


We consider Darboux transformations for operators of arbitrary order and construct the general theory of Bäcklund transformations based on the Lagrangian formalism. The dressing chain for the Boussinesq equation and its reduction are demonstrative examples for the suggested general theory. We also discuss the well-known Bäcklund transformations for classical soliton equations.


Soliton General Theory Arbitrary Order Lagrangian Formalism Boussinesq Equation 
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  1. 1.
    A. B. Shabat, Theor. Math. Phys., 121, 1397-1408 (1999).Google Scholar
  2. 2.
    A. P. Veselov and A. B. Shabat, Funct. Anal. Appl., 27, 81-96 (1993).Google Scholar
  3. 3.
    J. Weiss, J. Math. Phys., 28, 2025-2039 (1987).Google Scholar
  4. 4.
    X.-B. Hu and H.-W. Tam, Inverse Problems, 17, 319-327 (2001).Google Scholar
  5. 5.
    A. K. Svinin, Inverse Problems, 17, 307-318 (2001).Google Scholar
  6. 6.
    O. I. Bogoyavlenskii, Breaking Solitons: Nonlinear Integral Equations [in Russian], Nauka, Moscow (1991).Google Scholar
  7. 7.
    A. B. Mikhailov, A. B. Shabat, and R. I. Yamilov, Russ. Math. Surv., 42, 1-63 (1987).Google Scholar
  8. 8.
    V. E. Adler and A. B. Shabat, Theor. Math. Phys., 112, 935-948 (1997).Google Scholar
  9. 9.
    V. G. Marikhin, JETP Letters, 66, 705-709 (1997).Google Scholar
  10. 10.
    A. B. Shabat and R. I. Yamilov, Leningrad Math. J., 2, 377-400 (1990).Google Scholar
  11. 11.
    R. I. Yamilov, “Classification of Toda type scalar lattices,” in: Proc. 8th Intl. Workshop NEEDS'92 (V. Makhankov, I. Puzynin, and O. Rashaev, eds.), World Scientific, Singapore (1993), pp. 423-431.Google Scholar
  12. 12.
    V. E. Adler, Theor. Math. Phys., 124, 897-908 (2000).Google Scholar
  13. 13.
    E. K. Sklyanin, Funct. Anal. Appl., 16, 263-270 (1982).Google Scholar
  14. 14.
    V. E. Adler, Int. Math. Res. Notices, No. 1, 1-4 (1998).Google Scholar
  15. 15.
    R. Conte, “Exact solutions of nonlinear partial differential equations by singularity analysis,” in: Direct and Inverse Methods in Nonlinear Evolution Equations (CIME School, Cetraro, 5–12 September 1999, A. Greco, ed.), Springer, Berlin (2000); nlin.SI/0009024 (2000).Google Scholar
  16. 16.
    C. Rogers and S. Carillo, Physica Scripta, 36, 865-869 (1987).Google Scholar
  17. 17.
    M. C. Nucci, J. Phys. A, 22, 2897-2913 (1989).Google Scholar
  18. 18.
    J. Satsuma and D. J. Kaup, J. Phys. Soc. Japan, 43, 692-697 (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. E. Adler
    • 1
  • V. G. Marikhin
    • 2
  • A. B. Shabat
    • 2
  1. 1.Mathematical InstituteUfa Center, RASUfaRussia
  2. 2.Landau Institute for Theoretical Physics, RASMoscowRussia

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