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Integrability of Nonlinear Systems and Perturbation Theory

  • Chapter
What Is Integrability?

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

Abstract

The theory of so-called integrable Hamiltonian wave systems arose as a result of the inverse scattering method discovery by Gardner, Green, Kruskal and Miura [1] for the Korteveg-de Vries equation. This discovery was initiated by the pioneering numerical experiment by Kruskal and Zabusky [2]. After a pragmatic phase, which was devoted to finding new soliton equations, the theory became rather complicated. One of its branches may be called the “qualitative theory of infinite-dimensional Hamiltonian systems”, to which the results reviewed in this paper belong. We consider only Hamiltonian systems possessing Hamiltonians with a quadratic part which may be transformed in normal variables to the form

$$H_0 = \sum\limits_{\alpha = 1}^N {\omega _k ^{\left( \alpha \right)} a_k ^{\left( \alpha \right)} a_k ^{*\left( \alpha \right)} dk} $$
((1.1.1))

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References

  1. C.S. Gardner, J.M. Green, M.D. Kruskal, R.M. Miura: Phys. Lett. 19, 1095–1097 (1967).

    Article  MATH  Google Scholar 

  2. NJ. Zabusky, M.D. Kruskal: Phys. Rev. Lett. 15, 240–243 (1965).

    Article  ADS  MATH  Google Scholar 

  3. A. Poincaré: New methods of celestial mechanics, in Selected Works, Vol. I, II (Nauka, Moscow 1971), pp. 1–358.

    Google Scholar 

  4. V.E. Zakharov, E.I. Schulman: Physica D1, 191–202 (1980).

    MathSciNet  ADS  Google Scholar 

  5. V.E. Zakharov, E.I. Schulman: Dokl. Akad. Nauk 283, 1325–1328 (1985).

    Google Scholar 

  6. V.E. Zakharov, E.I. Schulman: Physica D29, 283–320 (1988).

    MathSciNet  ADS  Google Scholar 

  7. E.I. Schulman: Teor. Mat. Fizika 76, 88–99 (1988), in Russian.

    Google Scholar 

  8. H.H. Chen, Y.C. Lee, J.E. Lin: Physica Scripta 20, 490–492 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. H.H. Chen, Y.C. Lee, J.E. Lin: Physica D26, 165–170 (1987).

    MathSciNet  ADS  Google Scholar 

  10. V.E. Zakharov: “Integrable Systems in Multidimensional Spaces”, in Lect. Notes Phys. Vol. 153 (Springer, Berlin-Heidelberg 1983) pp. 190–216.

    Google Scholar 

  11. H. Umezava: The Quantum Field Theory (Inostr. Literatura, Moscow 1985), p. 380.

    Google Scholar 

  12. E.I. Schulman: Dokl. Acad. Nauk, 259, 579–781 (1981).

    Google Scholar 

  13. V.E. Zakharov, E.I. Schulman: Physica D4, 270–274 (1982).

    MathSciNet  ADS  Google Scholar 

  14. I.M. Krichever: Dokl. Akad. Nauk in press.

    Google Scholar 

  15. V.E. Zakharov: Integrable turbulence. Talk presented at the International Workshop “Nonlinear and Turbulent Processes in Physics”, Kiev, 1983.

    Google Scholar 

  16. A. Daveay, K. Stewartson: Proc. Roy. Soc. Lond. A338, 101–110 (1974).

    ADS  Google Scholar 

  17. H.H. Chen, J.E. Lin: “Constraints in the Kadomtsev-Petviashvili equation” — preprint N82-112, University of Maryland (1981).

    Google Scholar 

  18. A. Reiman, M. Semionov-Tian Shanskii: Proc. of LOMI Scientific Seminars 133, 212–227 (1984).

    Google Scholar 

  19. R.K. Bullogh, S.V. Manakov, ZJ. Jiang: Physica D18, 305–307 (1988).

    Google Scholar 

  20. E.I. Schulman: Teor. Mat. Fizika 56, 131–136 (1983).

    Google Scholar 

  21. E.P. Wigner: Symmetries and Reflections (Indiana University Press, Bloomington 1970).

    Google Scholar 

  22. V.E. Zakharov: “Kolmogorov Spectra in the Theory of Weak Turbulence”, in Basic Plasma Physics, ed. by M.N. Rosenbluth, R.Z. Sagdeev (North-Holland, Amsterdam 1984).

    Google Scholar 

  23. V.E. Zakharov: Izv. Vyssh. Uchebn. Zaved. Radiofiz, 17, 431–453 (1974).

    Google Scholar 

  24. V.E. Zakharov, A.M. Rubenchik: Prikl. Mat. Techn. Fiz. 5, 84–98 (1972).

    Google Scholar 

  25. A.P. Veselov, S.P. Novkov: Dokl. Akad. Nauk 279, 784–788 (1984).

    Google Scholar 

  26. V.I. Shrira: J. Nnl. Mech. 16, 129–138 (1982).

    MathSciNet  Google Scholar 

  27. V.E. Zakharov, E.A. Kuznetsov: Physica D18, 455–463 (1986).

    MathSciNet  ADS  Google Scholar 

  28. E.S. Benilov, S.P. Burtsev: Phys. Lett. A98, 256–258 (1983).

    MathSciNet  ADS  Google Scholar 

  29. D.D. Tskhakaia: Teor. Math. Fizica 77, in press.

    Google Scholar 

  30. A.X. Berkhoer, V.E. Zakharov: Zh. Eksp. Teor. Fiz. 58, 903–911 (1970).

    Google Scholar 

  31. S.M. Manakov: Zh. Eksp. Teor. Fiz. 65, 505–516 (1973).

    ADS  Google Scholar 

  32. V.E. Zakharov, A.B. Shabat: Funct. Anal. Appl. 8, 43–53 (1974).

    Google Scholar 

  33. V.G. Makhankov, O.K. Pashaev: “On Properties of the Nonlinear Schrödinger Equation with U(p,q) Symmetry”, JINR-Dubna preprint E2-81-70 (Dubna, USSR 1981).

    Google Scholar 

  34. V.E. Zakharov: Zh. Eksp. Teor. Fiz. 62, 1745–1759 (1972).

    Google Scholar 

  35. V.E. Zakharov, A.V. Mikhailov: Pisma Zh. Eksp. Teor. Fiz. 45, 279–282 (1987).

    Google Scholar 

  36. I.V. Cherednic: Teor. Mat. Fizika 47, 537–542 (1981).

    Google Scholar 

  37. A.P. Veselov: Dokl. Acad. Nauk 270, 1298–1300 (1983).

    MathSciNet  Google Scholar 

  38. V.Z. Khukhunashvili: Teor. Mat. Fizika 79, 180–184 (1989).

    MathSciNet  Google Scholar 

  39. A.P. Fordy, P.P. Kulish: Comm. Pure Appl. Math. 89, 427–443 (1983).

    MathSciNet  ADS  MATH  Google Scholar 

  40. S.V. Manakov: Physica D3, 420–427 (1981).

    MathSciNet  ADS  Google Scholar 

  41. W.E. Zakharov: Fund. Anal. Appl. 23 (1989), in press.

    Google Scholar 

  42. W.E. Zakharov, B.G. Konopelchenko: Comm. Math. Phys. 93, 483–509 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  43. A.S. Fokas, P.M. Santini: Stud. Appl. Math., 75, 179 (1986).

    MathSciNet  MATH  Google Scholar 

  44. A.S. Fokas, P.M. Santini: Comm. Math. Phys., 116, 449–474 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. P.M. Santini, A.S. Fokas: Comm. Math. Phys., 115, 375–419 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. A.Yu. Orlov, E.I. Schulman: “Additional Symmetries of Two-Dimensional Integrable Systems”, preprint Inst. Autom. Electrometry N217 (1985), Teor. Mat. Fizika 64, 323–327 (1985).

    Google Scholar 

  47. A.Yu. Orlov, E.I. Schulman: Lett. Math. Phys. 12, 171–179 (1986).

    Google Scholar 

  48. A.Yu. Orlov: “Vertex Operator, d-Problem, Symmetries, Variational Identifies and Hamiltonian Formalism for (2+1) Integrable Systems”, in Proc. Int. Workshop on Nonlinear and Turbulent Processes in Physics (1987), ed. by V.G. Bar’akhtar, V.M. Chernousenko, N.S. Erokhin, V.E. Zakharov (World Scientific, Singapore 1988), p. 116–134.

    Google Scholar 

  49. A.M. Balk, S.B. Nazarenko, V.E. Zakharov: Zh. Exp. Teor. Fiz. 97 (1990), in press.

    Google Scholar 

  50. E.I. Schulman, D.D. Tzakaya: Funkt. Anal. Pril. 25 (1991), in press.

    Google Scholar 

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Authors
  1. V. E. Zakharov
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  2. E. I. Schulman
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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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Zakharov, V.E., Schulman, E.I. (1991). Integrability of Nonlinear Systems and Perturbation Theory. 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_5

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  • DOI: https://doi.org/10.1007/978-3-642-88703-1_5

  • 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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