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Integrability

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What Is Integrability?

Part of the book series: Springer Series in Nonlinear Dynamics ((SSNONLINEAR))

Abstract

A comprehensive definition of the term “integrable” is proving to be elusive. Rather, use of this term invokes a variety of intuitive notions (and not infrequently, some lively debate) corresponding to a belief that integrable systems are in some sense “exactly soluble” and exhibit globally (i.e., for all initial conditions) “regular” solutions. In contrast, the term “nonintegrable” is, generally, taken to imply that a system cannot be “solved exactly” and that its solutions can behave in an “irregular” fashion. Here the notion of irregular behavior corresponds to dynamics that are very sensitive to initial conditions, with neighboring trajectories in the phase space locally diverging on the average at an exponential rate. This characteristic is measured by Lyapunov exponents. A system with at least one positive exponent will display irregular motion. In contrast, regular motion is associated with no positive exponents. Unfortunately, the definition of the Lyapunov exponents involves long time averages, their existence is only guaranteed for a limited set of situations and their values are difficult to compute both analytically and numerically. It is unlikely, therefore, that an algorithm which tests a given system for Lyapunov exponents will be a successful test for integrability.

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References

  1. H. Flaschka, A. C. Newell: Commun. Math. Phys. 76, 65–116 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. H. Flaschka, A. C. Newell: Physica 3 D 203–222 (1981)

    ADS  Google Scholar 

  3. H. Flaschka, A. C. Newell: Math Studies Vol. 61, ed. by A. Bishop, D. Campbell, B. Nicolaenko (North-Holland, Amsterdam 1982) 65–91.

    Google Scholar 

  4. M. Sato, T. Miwa, M. Jimbo: Physica 1D, 80 (1980); in Mathematical Problems in Theoretical Physics, ed. by K. Osterwalder, Lect. Notes Phys. Vol. 116 (Springer, New York 1980) 126; in Nuclear Spectroscopy, ed. by G.F. Bertsch, D. Kurath, Lect. Notes Phys. Vol. 119 (Springer, New York 1980).

    MathSciNet  ADS  Google Scholar 

  5. C. S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura: Phys. Rev. Lett. 19, 1095 (1967).

    Article  ADS  MATH  Google Scholar 

  6. C. S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura: Comm. Pure Appl. Math. 27, 97 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  7. V.E. Zakharov, L.D. Faddeev: Anal. Appl. 5, 280 (1971).

    Google Scholar 

  8. C. S. Gardner: J. Math. Phys. 12, 1548 (1971).

    Article  ADS  MATH  Google Scholar 

  9. M.D. Kruskal, N.J. Zabusky: Phys. Rev. Lett. 15, 240 (1965).

    Article  ADS  MATH  Google Scholar 

  10. S. Kovalevskaya: Acta Math. 12, 177 (1889).

    Article  MathSciNet  Google Scholar 

  11. S. Kovalevskaya: Acta Math. 14, 81 (1890).

    Article  MathSciNet  Google Scholar 

  12. A. C. Newell, M. Tabor, Y. Zeng: Physica, 12 D, 1 (1987).

    MathSciNet  ADS  Google Scholar 

  13. J. Weiss, M. Tabor, G. Carnevale: J. Math. Phys. 24, 522–526 (1983).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. J. Weiss: J. Math. Phys. 24, 1405–1413 (1983).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. J. Weiss: J. Math. Phys. 25, 13–24 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. J. Weiss: J. Math. Phys. 26, 258–269 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. J. Weiss: J. Math. Phys. 26, 2174–2180 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. D. V. Chudnovsky, G. V. Chudnovsky, M. Tabor: Phys. Lett. A97, 268–274 (1983).

    MathSciNet  ADS  Google Scholar 

  19. J.D. Gibbon, P. Radmore, M. Tabor, D. Wood: Stud. Appl. Math. 72, 39–63 (1985).

    MathSciNet  MATH  Google Scholar 

  20. J.G. Gibbon, M. Tabor: J. Math. Phys. 26, 1956–1960 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. M. Adler, P. van Moerbeke: Invent. Math. 67, 297–331 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. M. Adler, P. van Moerbeke: Commun. Math. Phys. 83, 83–106 (1982).

    Article  ADS  MATH  Google Scholar 

  23. L. Haine: Commun. Math. Phys. 94, 271 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  24. N. Ercolani, E. Siggia: “The Painlevé Property and Integrability”, Phys. Lett. A119, 112 (1986).

    MathSciNet  ADS  Google Scholar 

  25. J. D. Gibbon, A. C. Newell, M. Tabor, Y. B. Zeng: “Lax Pairs, Bäcklund Transformations and Special Solutions for Ordinary Differential Equations”, Nonlinearity 1, 481 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. R. Hirota: “Direct Methods in Soliton Theory”, in Solitons, ed. by R.K. Bullough, P. J. Caudrey, Topics Curr. Phys. Vol. 17 (Springer, New York 1980).

    Google Scholar 

  27. E. Date, M. Jimbo, M. Kashiwara, T. Miwa: “Transformation Groups for Soliton Equations”, in Proc. RIMS Symp. Nonlinear Integrable Systems — Classical and Quantum Theory, ed. by M. Jimbo, T. Miwa (World Scientific, Singapore 1983).

    Google Scholar 

  28. A.C. Newell: “Solitons in Mathematics and Physics”, Conf. Bd. Math. Soc. 43, Soc. Indust. Appl. Maths. (1985).

    Google Scholar 

  29. J. Weiss: J. Math. Phys. 25, 2226–2235 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. F. Cariello, M. Tabor: “Painlevé Expansions for Nonintegrable Evolution Equations” Physica 39 D, 77 (1989).

    MathSciNet  ADS  Google Scholar 

  31. J. D. Fourier, G. Levine, M. Tabor: “Singularity Clustering in the Duffing Oscillator”, J. Phys A 21, 33 (1988).

    ADS  Google Scholar 

  32. Y.T. Chang, M. Tabor, J. Weiss: J. Math. Phys. 23, 531–536 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. J. Weiss: Phys. Lett. A 105, 387 (1984).

    ADS  Google Scholar 

  34. M. Toda: Theory of Nonlinear Lattices, Springer Ser. Solid-State Sci. Vol. 20 (Springer, Berlin-Heildelberg 1981).

    Book  MATH  Google Scholar 

  35. M. Adler, P. van Moerbeke: Commun. Math. Phys. 83, 83 (1982).

    Article  ADS  MATH  Google Scholar 

  36. H. Yoshida: in Nonlinear Integrable Systems — Classical Theory and Quantum Theory, ed. by M. Jimbo, T. Miwa (World-Scientific, Singapor 1983) 273.

    Google Scholar 

  37. O.I. Bogoyavlenskii: Commun. Math. Phys. 51, 201 (1976).

    Article  ADS  Google Scholar 

  38. H. Flaschka, Y. Zeng: in preparation.

    Google Scholar 

  39. H. Flaschka: “The Toda Lattice in the Complex Domain”, preprint, Univ. of Arizona (1987).

    Google Scholar 

  40. M. Sato: RIMS Kokyuroku 439, 30 (1981).

    Google Scholar 

  41. G. Segal, G. Wilson: Publ. Math. Inst. Haute Etudes Scientifique 61, 5 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. F. Chang, M. Tabor, J. Weiss: J. Math. Phys. 23, 531 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Y.F. Chang, J.M. Greene, M. Tabor, J. Weiss: Physica 8 D, 183 (1983).

    MathSciNet  ADS  Google Scholar 

  44. J.D. Fourier, G. Levine, M. Tabor: J. Phys. A 21, 33 (1988).

    ADS  Google Scholar 

  45. M. Tabor, J. Weiss: Phys. Rev. A 24, 2157 (1981).

    ADS  Google Scholar 

  46. H. Segur: “Solitons and the Inverse Scattering Transform”, in Lectures given at the International School of Physics, “Enrico Fermi”, Varenna, Italy 7–9 July, 1980.

    Google Scholar 

  47. G. Levine, M. Tabor: “Integrating the Nonintegrable: Analytic Structure of the Lorenz System Revisited”, Physica 33 D, 189 (1988).

    MathSciNet  ADS  Google Scholar 

  48. M. Kus: J. Phys. A, L689 (1983).

    Google Scholar 

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Authors
  1. H. Flaschka
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  2. A. C. Newell
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  3. M. Tabor
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Editor information

Editors and Affiliations

  1. Landau Institute for Theoretical Physics, USSR Academy of Sciences, ul. Kosygina 2, SU-117334, Moscow, USSR

    Vladimir E. Zakharov

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© 1991 Springer-Verlag Berlin Heidelberg

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Flaschka, H., Newell, A.C., Tabor, M. (1991). Integrability. In: Zakharov, V.E. (eds) What Is Integrability?. Springer Series in Nonlinear Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88703-1_3

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  • DOI: https://doi.org/10.1007/978-3-642-88703-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-88705-5

  • Online ISBN: 978-3-642-88703-1

  • eBook Packages: Springer Book Archive

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