Soliton surfaces and their applications (soliton geometry from spectral problems)

  • Antoni Sym
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 239)


The paper contains a complete presentation of the ideas and results of the approach of soliton surfaces (manifolds). In this approach any n-dim. soliton system with a matrix real semi-simple Lie algebra g possesses its own geometry of n-dim. submanifolds of g. Various applications of this approach are discussed. A particular attention is paid to integrable classical string models.


Fundamental Form Reconstruction Problem Affine Space Fundamental Object Constant Sectional Curvature 
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.
    G.L. Lamb, Jr., from the letter to the author.Google Scholar
  2. 2.
    L.P. Eisenhart: A Treatise on the Differential Geometry of Curves and Surfaces (New York, N.Y., 1960)Google Scholar
  3. 3.
    L. Bianchi: Lezioni di geometria differenziale (Pisa, 1922)Google Scholar
  4. 4.
    R.L. Anderson and N.H. Ibragimov: Lie-Bäcklund transformations in applications(SIAM, Philadelphia 1979)Google Scholar
  5. 5.
    E.L. Ince: Ordinary Differential Equations(Dover, N.Y. 1956)Google Scholar
  6. 6.
    D. Levi, 0. Ragnisco and A. Sym: to be published in Il Nuovo CimentoGoogle Scholar
  7. 7.
    T.J. Willmore: Total curvature in Riemannian Geometry (Ellis Horwood Limited, Chichester, 1982)Google Scholar
  8. 8.
    L.P. Eisenhart: Riemannian Geometry (Princeton, N.J., 1949)Google Scholar
  9. 9.
    S.V. Manakov, S.P. Novikov, L.P. Pitaievsky and V.E. Zakharov: Tieoriya solitonov (Moscow, 1980)Google Scholar
  10. 10.
    M.J. Ablowitz and H. Segur: Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)Google Scholar
  11. 11a.
    A. Sym: Soliton theory is surface theory, preprint IFT-11-1981Google Scholar
  12. 11b.
    A. Sym: Lett.Nuovo Cimento 41, 353 (1984)Google Scholar
  13. 12.
    J. Harnad, Y. Saint-Aubin and S. Shnider: The soliton correlation matrix and the reduction problem for integrable systems, preprint, CRMA-1175, June 83Google Scholar
  14. 13.
    M. Bruschi, D. Levi, 0. Ragnisco and A.Sym: to be publishedGoogle Scholar
  15. 14.
    S.S. Chern and K. Tenenblat: J.Diff.Geometry 16, 347 (81)Google Scholar
  16. 15.
    R. Sasaki: Nucl.Phys. B 154, 343 (1979)CrossRefGoogle Scholar
  17. 16.
    A. Sym: to be publishedGoogle Scholar
  18. 17.
    P. Forgacs, Z. Horvath and L. Palla: Ann. of Phys. 136, 371 (81)CrossRefGoogle Scholar
  19. 18.
    F. Calogero and A. Degasperis: Spectral Transform and Solitons: tools to solve and investigate nonlinear evolution equations, v.1 (North-Holland Publ.Comp., 1982)Google Scholar
  20. 19.
    R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris: Solitons and Nonlinear Wave Equations (Academic Press, 1982)Google Scholar
  21. 20.
    V.E. Zakharov and S.V. Manakov: in Soviet Scientific Reviews, Sec.A v.1, 133 (1979)Google Scholar
  22. 21.
    F.D. Gakhov: Boundary problems (Moscow, 1977), in RussianGoogle Scholar
  23. 22.
    M.P. do Carmo: Differential Geometry of Curves and Surfaces (Prentice-Hall, Inc., 1976)Google Scholar
  24. 23.
    V.F. Kagan: Foundations of surface theory (Moscow-Leningrad, 1947), in RussianGoogle Scholar
  25. 24.
    S. Kobayashi and K. Nomizu: Foundations in differential geometry (Wiley-Interscience, 1963 and 1969).Google Scholar
  26. 25.
    Bang-yen Chen: Geometry of Submanifolds (M. Dekker Inc., 1973)Google Scholar
  27. 26.
    F. Lund and T. Regge: Phys.Rev. D 14, 1524 (1976)CrossRefGoogle Scholar
  28. 27.
    K. Tenenblat and C.L. Terng: Ann. of Math. 111, 477 (1980)Google Scholar
  29. 28.
    C.L. Terng: Ann of Math. 111, 491 (1980)Google Scholar
  30. 29.
    E. Cartan: Bull. Soc. Math. France 47, 125 (1919) and 48, 132 (1920)Google Scholar
  31. 30.
    H.T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover Publications, Inc., 1962)Google Scholar
  32. 31.
    P.G. Drazin: London Mathematical Society Lecture Note Series 85 (Cambridge University Press, 1983)Google Scholar
  33. 32.
    B.M. Barbashov, V.V. Nesterenko, A.M. Chervyakov: Teor. Mat. Fiz. 59, 209 (1984)Google Scholar
  34. 33.
    V.V. Nesterenko: P2-80-342, Dubna preprintGoogle Scholar
  35. 34.
    M. Gürses and Y. Nutku: J. Math. Phys. 22, 1393 (1981)CrossRefGoogle Scholar
  36. 35.
    J. Eells and J.H. Sampson: Amer. J. Math. 86, 109 (1964)Google Scholar
  37. 36.
    A.M. Perelomov: Physica 4D, 1 (1981)Google Scholar
  38. 37.
    A.O. Barut and R. Raczka: Theory of Group Representations and Applications (PWN, Warsaw 1977)Google Scholar
  39. 38.
    V.L. Golo and B.A. Putko: Teor. Mat. Fiz: 45, 19 (1980)Google Scholar
  40. 39.
    B.A. Putko: Teor. Mat. Fiz. 50, 108 (1982)Google Scholar
  41. 40.
    F. Lund: Ann. Phys. 115, 251 (1978)CrossRefGoogle Scholar
  42. 41.
    A. Sym: Lett. Nuovo Cim. 36, 307 (1983)Google Scholar
  43. 42.
    A. Sym: Lett. Nuovo Cim. 41, 353 (1984) and references therein quoted.Google Scholar
  44. 43.
    M. Kalb and P. Ramond: Phys. Rev. D9, 2273 (1974)Google Scholar
  45. 44.
    D. Levi, A. Sym and S. Wojciechowski: Phys. Lett. A 94, 408 (1983CrossRefGoogle Scholar
  46. 45.
    A. Sym: Lett. Nuovo Cim. 40, 225 (1984)Google Scholar
  47. 46.
    A. Sym: Lett. Nuovo Cim. 41, 33 (1984)Google Scholar
  48. 47.
    D. Levi, O. Ragnisco and A. Sym: to be published in Il Nuovo Cimento and references therein quotedGoogle Scholar
  49. 48.
    G. Neugebauer and R. Meinel: Phys.Lett. 100A, 467 (1984)Google Scholar
  50. 49.
    A. Sym: to be publishedGoogle Scholar
  51. 50.
    J. Cieslinski and A. Sym: to be publishedGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Antoni Sym
    • 1
  1. 1.Institute of Theoretical Physics of Warsaw UniversityWarsawPoland

Personalised recommendations