, Volume 61, Issue 1, pp 99–107 | Cite as

Canonical structure of evolution equations with non-linear dispersive terms

  • B. Talukdar
  • J. Shamanna
  • S. Ghosh


The inverse problem of the variational calculus for evolution equations characterized by non-linear dispersive terms is analysed with a view to clarify why such a system does not follow from Lagrangians. Conditions are derived under which one could construct similar equations which admit a Lagrangian representation. It is shown that the system of equations thus obtained can be Hamiltonized by making use of the Dirac’s theory of constraints. The specific results presented refer to the third- and fifth-order equations of the so-called distinguished subclass.


Evolution equations non-linear dispersive terms Lagrangian systems Hamiltonian structure 


47.20.Ky 42.81.Dp 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F Calogero and A Degesparis,Spectral transform and solitons (North Holland Publ. Co., NY, 1982) vol. 1MATHGoogle Scholar
  2. [2]
    P Rosenau and J M Hyman,Phys. Rev. Lett. 70, 564 (1993)MATHCrossRefADSGoogle Scholar
  3. [3]
    B Dey and A Khare,J. Phys. 33, 5335 (2000)MATHCrossRefMathSciNetADSGoogle Scholar
  4. [4]
    S A Hojman and L C Shepley,J. Math. Phys. 32, 142 (1990)CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    G Darboux, Ivieme partie (Gauthier-Villars Paris, 1891)Google Scholar
  6. [5]a
    G Lopez,Ann. Phys. (NY) 251, 363 (1996)MATHCrossRefADSGoogle Scholar
  7. [6]
    R M Santili,Foundations of theoretical mechanics (Springer-Verlag, NY, 1978) vol. 1Google Scholar
  8. [7]
    P J Olver,Applications of Lie groups to differential equations (Springer-Verlag, NY, 1993)MATHGoogle Scholar
  9. [8]
    E C G Sudarshan and N Mukunda,Classical dynamics: A modern perspective (John-Wiley and Sons Inc., NY, 1974)MATHGoogle Scholar
  10. [9]
    P J Olver,J. Math. Phys. 27, 2495 (1986)MATHCrossRefADSMathSciNetGoogle Scholar
  11. [10]
    Y Nutku,J. Phys. A29, 3257 (1996)ADSMathSciNetGoogle Scholar
  12. [11]
    R M Santili,Foundations of theoretical mechanics (Springer-Verlag, NY, 1978) vol. IIGoogle Scholar
  13. [12]
    P Rosenau,Phys. Lett. A230, 305 (1997)ADSMathSciNetGoogle Scholar
  14. [13]
    V G Karmanov,Mathematical programming (Mir Publishers, Moscow, 1989)MATHGoogle Scholar
  15. [14]
    F J deUrries and J Julve,J. Phys. A31, 6949 (1998)ADSMathSciNetGoogle Scholar
  16. [15]
    G B Whitham,Proc. R. Soc. (London) Ser. A283, 238 (1965)ADSMathSciNetCrossRefGoogle Scholar
  17. [16]
    P A M Dirac,Lectures on quantum mechanics (Belfer Graduate School Monograph Series No. 2, Yeshiva University, NY, 1964)Google Scholar
  18. [17]
    V E Zakharov and L D Faddeev,Funct. Anal. Appl. 5, 18 (1971)Google Scholar
  19. [17]a
    C S Gardner,J. Math. Phys. 12, 1548 (1971)MATHCrossRefADSGoogle Scholar
  20. [18]
    L Faddeev and R Jackiw,Phys. Rev. Lett. 60, 1692 (1988)CrossRefADSMathSciNetMATHGoogle Scholar
  21. [19]
    H Montani,Int. J. Mod. Phys. A8, 4319 (1993)ADSMathSciNetGoogle Scholar
  22. [19]a
    M M Horta-Barreira and C Wotzasek,Phys. Rev. D45 1410 (1992)ADSMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2003

Authors and Affiliations

  • B. Talukdar
    • 1
  • J. Shamanna
    • 1
  • S. Ghosh
    • 1
  1. 1.Department of PhysicsVisva-Bharati UniversitySantiniketanIndia

Personalised recommendations