Skip to main content

Integrability and Nijenhuis Tensors

  • Chapter
Integrable Hamiltonian Hierarchies

Part of the book series: Lecture Notes in Physics ((LNP,volume 748))

  • 1326 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  2. V. E. Zakharov and L. D. Faddeev. Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl., 5(4):280–287, 1971.

    Article  MATH  Google Scholar 

  3. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. On the Phase Manifold Geometry of Integrable Nonlinear Field Theory. Preprint IFUSA, 1982.

    Google Scholar 

  4. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. A new characterization of complete integrable systems. Nuovo Cimento B, 83:97–112, 1984.

    Article  ADS  Google Scholar 

  5. D. J. Kaup. Closure of the squared Zakharov–Shabat eigenstates. J. Math. Anal. Appl., 54(3):849–864, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. J. Olver. Evolution equations possessing infinitely many symmetries. J. Math. Phys., 18:1212, 1977.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Magri, F.: A geometrical approach to the nonlinear solvable equations. In: Boiti, M., Pempinelli, F., Soliani, G. (eds.) Nonlinear Evolution Equations and Dynamical Systems: Proceedings of the Meeting Held at the University of Lecce June 20–23, 1979. Lect. Notes Phys. 120, 233–263 (1980)

    Google Scholar 

  8. G. Vilasi. On the hamiltonian structures of the Korteweg-de Vries and sine-Gordon theories. Phys. Lett. B, 94(2):195–198, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  9. S. De Filippo, G. Marmo, and G. Vilasi. A geometrical setting for the Lax representation. Phys. Lett. B, 117(6):418–422, 1983.

    Google Scholar 

  10. S. De Filippo, M. Salerno, and G. Vilasi. A geometrical approach to the integrability of soliton equations. Lett. Math. Phys., 9:85–91, 1985.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. E. Hopf. The partial differential equation u t + uu x = u xx . Comm. Pure Appl. Math., 3:201–230, 1950.

    Google Scholar 

  12. J. D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math., 9(3):225–236, 1951.

    MATH  MathSciNet  Google Scholar 

  13. W. F. Ames. Nonlinear Partial Differential Equations in Engineering. Academic Press, New York-London edition, 1965.

    Google Scholar 

  14. G. Vilasi. Phase manifold geometry of burgers hierarchy. Lett. Nuovo Cimento, 37(3):105–109, 1985.

    MathSciNet  Google Scholar 

  15. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. Phase manifold geometry of burgers hierarchy. Lett. Nuovo Cimento, 37(3):105–109, 1983.

    Google Scholar 

  16. A. S. Mishchenko and A. T. Fomenko. Generalized Liouville method of integration of Hamiltonian systems. Funct. Anal. Appl., 12(2):113–121, 1978.

    Article  MATH  Google Scholar 

  17. M. R. Adams and T. Ratiu. The three-point vortex problem: Commutative and noncommutative integrability. Contemp. Math., 81:245–257, 1988.

    MathSciNet  Google Scholar 

  18. V. V. Trofimov and A. T. Fomenko. Algebra and Geometry of the Integrable Hamiltonian Differential Equations. Factorial, Minsk, 1995.

    Google Scholar 

  19. A. T. Fomenko. Symplectic Geometry. Advanced Studies in Contemporary Mathematics. Gordon & Breach Publishers, Luxembourg, 1995.

    Google Scholar 

  20. F. Fassò and T. Ratiu. Compatibility of symplectic structures adapted to noncommutatively integrable systems. J. Geom. Phys., 27:199–220, 1998.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. O. I. Bogoyavlenskij. Extended integrability and Bi-Hamiltonian systems. Commun. Math. Phys., 196(1):19–51, 1998.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. O. I. Bogoyavlenskij. Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures. Commun. Math. Phys., 180(3):529–586, 1996.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. O. I. Bogoyavlenskij. Theory of tensor invariants of integrable Hamiltonian systems. II. Theorem on symmetries and its applications. Commun. Math. Phys., 184(2):301–365, 1997.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. B. Cordani. The Kepler Problem:Group Theoretical Aspects, Regularization and Quantization, with Applications to the Study of Perturbations. Birkhauser Verlag, Boston, MA, 2003.

    Google Scholar 

  25. G. Marmo and G. Vilasi. When do recursion operators generate new conservation laws? Phys. Lett. B, 277(1–2):137–140, 1992.

    ADS  MathSciNet  Google Scholar 

  26. F. Magri and C. Morosi. A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds. Quaderni del Dipartimento di Matematica, Università di Milano, 1984.

    Google Scholar 

  27. G. Magnano and F. Magri. Poisson–Nijenhuis structures and Sato hierarchy. Rev. Math. Phys., 3(4):403–466, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  28. Y. Kosmann-Schwarzbach and F. Magri. Lax–Nijenhuis operators for integrable systems. J. Math. Phys., 37:6173–6197, 1996.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Y. Kosmann-Schwarzbach. The Lie bialgebroid of a Poisson–Nijenhuis manifold. Lett. Math. Phys., 38(4):421–428, 1996.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. V. G. Drinfeld and V. V. Sokolov. Lie Algebras and Korteweg-de Vries Type Equations. VINITI Series: Contemporary problems of mathematics. Recent developments. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985.

    Google Scholar 

  31. L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  32. F. Magri, C. Morosi, and O. Ragnisco. Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications. Commun. Math. Phys., 99(1):115–140, 1985.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Magri, F., Falqui, G., Pedroni. M.: The method of Poisson pairs in the theory of nonlinear PDEs. Direct and Inverse Methods in Nonlinear Evolution Equations. Lect. Notes Phys. 632, 85–136. Springer-Verlag, Berlin (2003)

    ADS  MathSciNet  Google Scholar 

  34. I. M. Gel’fand and I. Y. Dorfman. Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl., 13(4):248–262, 1979.

    Article  MathSciNet  Google Scholar 

  35. I. M. Gel’fand and I. Y. Dorfman. The schouten bracket and hamiltonian operators. Funct. Anal. Appl., 14(3):223–226, 1980.

    Article  MathSciNet  Google Scholar 

  36. C. Godbillion. Géométrie différentielle et méchanique analytique. Hermann, Paris, 1969.

    Google Scholar 

  37. B. Florko and A. B. Yanovski. On Magri’s theorem for complete integrability. Communication JINR (Joint Institute for Nuclear Research), Dubna, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). Integrability and Nijenhuis Tensors. In: Gerdjikov, V., Vilasi, G., Yanovski, A. (eds) Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77054-1_14

Download citation

Publish with us

Policies and ethics