On the Role of Virasoro, Kac-Moody Algebra and Conformal Invariance in Soliton Hierarchies

  • A. Roy Chowdhury
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

The implications of infinite dimensional Lie algebra, Virasoro and Kac-Moody algebras in the study of completely integrable partial differential equations, are reviewed. Intimate relations are shown to exist between prolongation algebra, Lie-Bäcklund algebra and such infinite dimensional algebras. The importance of conformal invariance is emphasized and some new aspects of complete integrability are exhibited. Lastly the relation between bi-Hamiltonian structure and conformal invariance is explained.

Keywords

Manifold Soliton Culmination 

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References

  1. [1]
    E. K. Bullough and P. J. Caudrey, eds., Solitons, Current Topics in Physics (Springer-Verlag, Berlin, New York, 1981); M. Lakshmanan, ed., Solitons: Introduction and Applications ( Springer-Verlag, Heidelberg, 1987 ).Google Scholar
  2. [2]
    M. J. Ablowitz, A. Ramani, H. Segur, Lett. Nuovo Cimento 23 (1978) 333; J. Math. Phys. 21 (1980) 715, 1006; J. Weiss, J. Math. Phys. 24 (1983) 1405; 25 (1984) 13; A. Roy Chowdhury and P. K. Chanda, J. Math. Phys. 27 (1987) 707; P. K. Chanda and A. Roy Chowdhury, J. Math. Phys. 29 (1988) 843; R. Sahadevan and M. Lakshmanan, Phys. Rev. A31 (1985) 861; Phys. Lett. 301A (1984) 189.MathSciNetCrossRefGoogle Scholar
  3. [3]
    G. L. Lamb, Jr., Elements of Soliton Theory ( John Wiley, New York, 1980 ).MATHGoogle Scholar
  4. [4]
    D. Sattinger, Stud. Appl Math. 72 (1983) 65;S,Olefsson, J. Phys. A22 (1989) 157; A. Roy Chowdhury and S. Roy, J. Math. Phys. 27 (1986) 2464.MathSciNetGoogle Scholar
  5. [5]
    P. Goddard and E. Olive, Int. J. Mod. Phys. A1 (1986) 303; V. Kac, Infinite Dimensional Lie Algebra ( Birkhauser, Boston, 1983 ).MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    H. D. Wahlquist and F. B. Estabrook, J. Math. Phys. 16 (1975) 1.MathSciNetADSMATHCrossRefGoogle Scholar
  7. [7]
    C. Hoenselares, Prog. Theo. Phys. 74 (1985) 645.ADSCrossRefGoogle Scholar
  8. [8]
    F. B. Estabrook and H. D. Wahlquist, J. Math. Phys. 17 (1976) 1293.MathSciNetADSMATHCrossRefGoogle Scholar
  9. [9]
    H. N. Van Eck, Proc. Netherland Acad. Sci. A86 (1983) 149.Google Scholar
  10. [10]
    S. Roy and A. Roy Chowdhury, Int. J. Theo. Phys. 28 (1989) 845.MATHCrossRefGoogle Scholar
  11. [11]
    G. Bhattacharya and H. Bohr, ICTP Preprint 81/82 (13).Google Scholar
  12. [12]
    L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Soliton ( Springer-Verlag, Berlin, 1988 ).Google Scholar
  13. [13]
    H. Eichenherr, CERN Preprint, TH 3491.Google Scholar
  14. [14]
    K. Ueno and Y. Nakamura, Phys. Lett. 109B (1982) 273.MathSciNetCrossRefGoogle Scholar
  15. [15]
    A. Mikailov, Physica 3D (1981) 73.Google Scholar
  16. [16]
    H. Bohr and A. Roy Chowdhury, J. Phys. A18 (1985) 3423.MathSciNetADSMATHGoogle Scholar
  17. [17]
    J. L. Gervais, Phys. Lett. 1608 (1985) 277; J. L. Gervais and A. Neveu, Nucl. Phys. B209 (1982) 125.MathSciNetGoogle Scholar
  18. [18]
    C. A. Laberge and P. Matheiu, Phys. Lett. B215 (1988) 718; P. Matheiu, J. Math. Phys. 29 (1988) 2499.MathSciNetCrossRefGoogle Scholar
  19. [19]
    T. Bakas, Nucl. Phys. B302 (1988) 189; E. Witten, Comm. Math. Phys. 114 (1988) 1;MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • A. Roy Chowdhury
    • 1
  1. 1.High Energy Physics Division, Department of PhysicsJadavpur UniversityCalcuttaIndia

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