Hamiltonian Structures of Soliton Equations via Constrained Variational Calculus

  • Tu Gui-zhang
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)


We prove the following trace identity
$$\left( {\delta /\delta {u_\alpha }} \right)Tr\left( {K\partial U/\partial n} \right) = \left( {\partial {\mkern 1mu} /{\mkern 1mu} \partial n} \right)Tr\left( {K\partial U/{\mkern 1mu} \partial {u_\alpha }} \right),$$
where K satisfies the equation Kx=[U,K] and U=U(uα,n). We show that a number of known formulae, obtained previously by complicated calculation, are the special cases of this remarkable identity.


Spectral Problem Adjoint Representation Recurrence Formula Hamiltonian Structure Variational Calculus 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Tu G.Z., J. Eng. Math., 1 (1984), 7. (in Chinese)Google Scholar
  2. 2.
    BOITI M., PEMPINELLT F. and TU G.Z., Nuovo Cimento 79B (1984), 231.CrossRefGoogle Scholar
  3. 3.
    TU. G.Z., J.Math. Anal. Appl., 94 (1983), 348.CrossRefGoogle Scholar
  4. 4.
    GELFAND I.M. and DIKII L.A., Funct. Anal. Appl. 10 (1976), 13. (in Russian).Google Scholar
  5. 5.
    ADLER M., Invent. Math., 50 (1979), 219.CrossRefGoogle Scholar
  6. 6.
    GUERRERO G.F., J. Math. Phys., 23 (1982), 211.CrossRefGoogle Scholar
  7. 7.
    TU G.Z., Comm. Math. Phys., 77 (1980), 289.Google Scholar
  8. 8.
    GELFAND I.M. and DORFIMAN I.YA., Current problems of mathematical physics and of numerical mathematics, Moskva (1982), 102. (in Russian).Google Scholar
  9. 9.
    GELFAND I.M. and DIKII L.A., Uspekhi Nauk Matem., 30(1975), 67. (in Russian)Google Scholar
  10. 10.
    YUSIN B.V., ibid, 33(1978), 233. (in Russian).Google Scholar
  11. 11.
    ALBERTY T.M., KOIKAWA T. and SASAKI R., Physica 5D (1982), 43.Google Scholar
  12. 12.
    BOITI M. and TU G.Z., Nuovo Cimento, 71B (1982), 253.CrossRefGoogle Scholar
  13. 13.
    TU G.Z., Sci. Exploration, 2 (1982), 85.Google Scholar
  14. 14.
    KAUP D.J. and NEWELL A.C., J.Math. Phys., 19 (1978), 798.CrossRefGoogle Scholar
  15. 15.
    HELGASON S., Differential Geometry, Lie groups and symmetric spaces, Academic Press: New York, 1978.Google Scholar
  16. 16.
    WILSON G., Ergod. Th. Dynam. Sys., 1 (1981 ), 361.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Tu Gui-zhang
    • 1
  1. 1.Computing Center of Chinese Academy of SciencesBeijingPR China

Personalised recommendations