Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 275450, 8 pages
http://dx.doi.org/10.1155/2014/275450
Research Article

Consecutive Rosochatius Deformations of the Garnier System and the Hénon-Heiles System

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

Received 8 January 2014; Accepted 20 February 2014; Published 31 March 2014

Academic Editor: Weiguo Rui

Copyright © 2014 Baoqiang Xia and Ruguang Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Rosochatius, “Uber die Bewegung eines Punktes,” [Ph.D. dissertation], University of Gotingen, 1877. View at Google Scholar
  2. C. Neumann, “De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur,” Journal für die Reine und Angewandte Mathematik, vol. 56, pp. 46–63, 1859. View at Publisher · View at Google Scholar
  3. J. Moser, “Geometry of quadrics and spectral theory,” in The Chern Symposium 1979, pp. 147–188, Springer, New York, NY, USA, 1980. View at Publisher · View at Google Scholar
  4. M. R. Adams, J. Harnad, and E. Previato, “Isospectral hamiltonian flows in finite and infinite dimensions. I: generalized Moser systems and moment maps into loop algebras,” Communications in Mathematical Physics, vol. 117, no. 3, pp. 451–500, 1988. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Harnad and P. Winternitz, “Classical and quantum integrable systems in 263-1263-1263-1 and separation of variables,” Communications in Mathematical Physics, vol. 172, no. 2, pp. 263–285, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. C. Bartocci, G. Falqui, and M. Pedroni, “A geometric approach to the separability of the Neumann-Rosochatius system,” Differential Geometry and Its Application, vol. 21, no. 3, pp. 349–360, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. C.-W. Cao and B.-Q. Xia, “From Rosochatius system to KdV equation,” Communications in Theoretical Physics, vol. 54, no. 4, pp. 619–624, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. S. Wojciechowski, “Integrability of one particle in a perturbed central quartic potential,” Physica Scripta, vol. 31, no. 6, pp. 433–438, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. Tondo, “On the integrability of stationary and restricted flows of the KdV hierarchy,” Journal of Physics A, vol. 28, no. 17, pp. 5097–5115, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Kubo, W. Ogura, T. Saito, and Y. Yasui, “The Gauss-Knörrer map for the Rosochatius dynamical system,” Physics Letters A, vol. 251, no. 1, pp. 6–12, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. B. Suris, “Discrete-time analogues of some nonlinear oscillators in the inverse-square potential,” Journal of Physics A, vol. 27, no. 24, pp. 8161–8169, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. A. N. W. Hone, V. B. Kuznetsov, and O. Ragnisco, “Bäcklund transformations for many-body systems related to KdV,” Journal of Physics A, vol. 32, no. 27, pp. L299–L306, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. Zhou, “Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations,” Journal of Mathematical Physics, vol. 48, no. 5, Article ID 053509, 23 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. Xia and R. Zhou, “Integrable deformations of integrable symplectic maps,” Physics Letters A, vol. 373, no. 47, pp. 4360–4367, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. Y. Q. Yao and Y. B. Zeng, “Integrable rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources,” Journal of Physics A, vol. 41, no. 29, Article ID 295205, 2008. View at Publisher · View at Google Scholar
  16. H. Dimov and R. C. Rashkov, “A note on spin chain/string duality,” International Journal of Modern Physics A, vol. 20, no. 18, pp. 4337–4353, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. Bozhilov, “Neumann and Neumann-Rosochatius integrable systems from membranes on Ad S4 × S7,” Journal of High Energy Physics, no. 8, article 073, 21 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  18. N. A. Kostov, “Quasi-periodic solutions of the integrable dynamical systems related to Hill's equation,” Letters in Mathematical Physics A, vol. 17, no. 2, pp. 95–108, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, and N. A. Kostov, “Quasi-periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type,” Proceedings of the Royal Society of London A, vol. 456, no. 2001, pp. 2263–2281, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Yao and Y. Zeng, “The Bi-hamiltonian structure and new solutions of KdV6 equation,” Letters in Mathematical Physics, vol. 86, no. 2-3, pp. 193–208, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. B. Q. Xia and R. G. Zhou, “Consecutive Rosochatius deformations of the Neumann system,” Journal of Mathematical Physics, vol. 54, no. 10, Article ID 103514, 2013. View at Publisher · View at Google Scholar
  22. V. B. Kuznetsov, “Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet,” Journal of Mathematical Physics, vol. 33, no. 9, pp. 3240–3254, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. C. Eilbeck, V. Z. Ènolskiĭ, V. B. Kuznetsov, and A. V. Tsiganov, “Linear r-matrix algebra for classical separable systems,” Journal of Physics A, vol. 27, no. 2, pp. 567–578, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  24. O. Babelon and C.-M. Viallet, “Hamiltonian structures and Lax equations,” Physics Letters B, vol. 237, no. 3-4, pp. 411–416, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Z. Qiao, “Generalized r-matrix structure and algebro-geometric solution for integrable system,” Reviews in Mathematical Physics A, vol. 13, no. 5, pp. 545–586, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  26. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, Germany, 1978. View at MathSciNet
  27. R. Garnier, “Sur une classe de systèmes différentiels abéliens déduits de la théorie des équations linéaires,” Rendiconti del Circolo Matematico di Palermo, vol. 43, no. 1, pp. 155–191, 1919. View at Publisher · View at Google Scholar
  28. C. W. Cao and X. G. Geng, “Classical integrable systems generated through nonlinearization of eigenvalue problems,” in Nonlinear Physics, C. H. Gu, Y. S. Li, and G. Z. Tu, Eds., Research Reports in Physics, pp. 66–78, Springer, Berlin, Germany, 1990. View at Publisher · View at Google Scholar
  29. B. Xia, “A hierarchy of Garnier-Rosochatius systems,” Journal of Mathematical Physics, vol. 52, no. 6, Article ID 063506, 11 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. M. Hénon and C. Heiles, “The applicability of the third integral of motion: some numerical experiments,” Astronomical Journal, vol. 69, p. 73, 1964. View at Publisher · View at Google Scholar
  31. Y. F. Chang, M. Tabor, and J. Weiss, “Analytic structure of the Hénon-Heiles Hamiltonian in integrable and nonintegrable regimes,” Journal of Mathematical Physics, vol. 23, no. 4, pp. 531–538, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. Antonowicz and S. Rauch-Wojciechowski, “Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions,” Physics Letters A, vol. 163, no. 3, pp. 167–172, 1992. View at Publisher · View at Google Scholar · View at MathSciNet