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

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Book cover Geometric Aspects of the Einstein Equations and Integrable Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 239))

Abstract

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.

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References

  1. G.L. Lamb, Jr., from the letter to the author.

    Google Scholar 

  2. L.P. Eisenhart: A Treatise on the Differential Geometry of Curves and Surfaces (New York, N.Y., 1960)

    Google Scholar 

  3. L. Bianchi: Lezioni di geometria differenziale (Pisa, 1922)

    Google Scholar 

  4. R.L. Anderson and N.H. Ibragimov: Lie-Bäcklund transformations in applications(SIAM, Philadelphia 1979)

    Google Scholar 

  5. E.L. Ince: Ordinary Differential Equations(Dover, N.Y. 1956)

    Google Scholar 

  6. D. Levi, 0. Ragnisco and A. Sym: to be published in Il Nuovo Cimento

    Google Scholar 

  7. T.J. Willmore: Total curvature in Riemannian Geometry (Ellis Horwood Limited, Chichester, 1982)

    Google Scholar 

  8. L.P. Eisenhart: Riemannian Geometry (Princeton, N.J., 1949)

    Google Scholar 

  9. S.V. Manakov, S.P. Novikov, L.P. Pitaievsky and V.E. Zakharov: Tieoriya solitonov (Moscow, 1980)

    Google Scholar 

  10. M.J. Ablowitz and H. Segur: Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)

    Google Scholar 

  11. A. Sym: Soliton theory is surface theory, preprint IFT-11-1981

    Google Scholar 

  12. A. Sym: Lett.Nuovo Cimento 41, 353 (1984)

    Google Scholar 

  13. J. Harnad, Y. Saint-Aubin and S. Shnider: The soliton correlation matrix and the reduction problem for integrable systems, preprint, CRMA-1175, June 83

    Google Scholar 

  14. M. Bruschi, D. Levi, 0. Ragnisco and A.Sym: to be published

    Google Scholar 

  15. S.S. Chern and K. Tenenblat: J.Diff.Geometry 16, 347 (81)

    Google Scholar 

  16. R. Sasaki: Nucl.Phys. B 154, 343 (1979)

    Article  Google Scholar 

  17. A. Sym: to be published

    Google Scholar 

  18. P. Forgacs, Z. Horvath and L. Palla: Ann. of Phys. 136, 371 (81)

    Article  Google Scholar 

  19. 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. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris: Solitons and Nonlinear Wave Equations (Academic Press, 1982)

    Google Scholar 

  21. V.E. Zakharov and S.V. Manakov: in Soviet Scientific Reviews, Sec.A v.1, 133 (1979)

    Google Scholar 

  22. F.D. Gakhov: Boundary problems (Moscow, 1977), in Russian

    Google Scholar 

  23. M.P. do Carmo: Differential Geometry of Curves and Surfaces (Prentice-Hall, Inc., 1976)

    Google Scholar 

  24. V.F. Kagan: Foundations of surface theory (Moscow-Leningrad, 1947), in Russian

    Google Scholar 

  25. S. Kobayashi and K. Nomizu: Foundations in differential geometry (Wiley-Interscience, 1963 and 1969).

    Google Scholar 

  26. Bang-yen Chen: Geometry of Submanifolds (M. Dekker Inc., 1973)

    Google Scholar 

  27. F. Lund and T. Regge: Phys.Rev. D 14, 1524 (1976)

    Article  Google Scholar 

  28. K. Tenenblat and C.L. Terng: Ann. of Math. 111, 477 (1980)

    Google Scholar 

  29. C.L. Terng: Ann of Math. 111, 491 (1980)

    Google Scholar 

  30. E. Cartan: Bull. Soc. Math. France 47, 125 (1919) and 48, 132 (1920)

    Google Scholar 

  31. H.T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover Publications, Inc., 1962)

    Google Scholar 

  32. P.G. Drazin: London Mathematical Society Lecture Note Series 85 (Cambridge University Press, 1983)

    Google Scholar 

  33. B.M. Barbashov, V.V. Nesterenko, A.M. Chervyakov: Teor. Mat. Fiz. 59, 209 (1984)

    Google Scholar 

  34. V.V. Nesterenko: P2-80-342, Dubna preprint

    Google Scholar 

  35. M. Gürses and Y. Nutku: J. Math. Phys. 22, 1393 (1981)

    Article  Google Scholar 

  36. J. Eells and J.H. Sampson: Amer. J. Math. 86, 109 (1964)

    Google Scholar 

  37. A.M. Perelomov: Physica 4D, 1 (1981)

    Google Scholar 

  38. A.O. Barut and R. Raczka: Theory of Group Representations and Applications (PWN, Warsaw 1977)

    Google Scholar 

  39. V.L. Golo and B.A. Putko: Teor. Mat. Fiz: 45, 19 (1980)

    Google Scholar 

  40. B.A. Putko: Teor. Mat. Fiz. 50, 108 (1982)

    Google Scholar 

  41. F. Lund: Ann. Phys. 115, 251 (1978)

    Article  Google Scholar 

  42. A. Sym: Lett. Nuovo Cim. 36, 307 (1983)

    Google Scholar 

  43. A. Sym: Lett. Nuovo Cim. 41, 353 (1984) and references therein quoted.

    Google Scholar 

  44. M. Kalb and P. Ramond: Phys. Rev. D9, 2273 (1974)

    Google Scholar 

  45. D. Levi, A. Sym and S. Wojciechowski: Phys. Lett. A 94, 408 (1983

    Article  Google Scholar 

  46. A. Sym: Lett. Nuovo Cim. 40, 225 (1984)

    Google Scholar 

  47. A. Sym: Lett. Nuovo Cim. 41, 33 (1984)

    Google Scholar 

  48. D. Levi, O. Ragnisco and A. Sym: to be published in Il Nuovo Cimento and references therein quoted

    Google Scholar 

  49. G. Neugebauer and R. Meinel: Phys.Lett. 100A, 467 (1984)

    Google Scholar 

  50. A. Sym: to be published

    Google Scholar 

  51. J. Cieslinski and A. Sym: to be published

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Theoretical Physics of Warsaw University, ul. Hoia 69, 00-681, Warsaw, Poland

    Antoni Sym

Authors
  1. Antoni Sym
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Editor information

R. Martini

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© 1985 Springer-Verlag

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Sym, A. (1985). Soliton surfaces and their applications (soliton geometry from spectral problems). In: Martini, R. (eds) Geometric Aspects of the Einstein Equations and Integrable Systems. Lecture Notes in Physics, vol 239. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16039-6_6

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  • DOI: https://doi.org/10.1007/3-540-16039-6_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16039-7

  • Online ISBN: 978-3-540-39713-7

  • eBook Packages: Springer Book Archive

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