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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 272904, 14 pages
http://dx.doi.org/10.1155/2012/272904
Research Article

Nonlinear Bi-Integrable Couplings of Multicomponent Guo Hierarchy with Self-Consistent Sources

1Information School, Shandong University of Science and Technology, Qingdao 266590, China
2Institute of Oceanology, China Academy of Sciences, Qingdao 266071, China
3Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China
4Department of Mathematics, Zaozhuang College, Zaozhuang 277160, China

Received 4 October 2012; Accepted 5 December 2012

Academic Editor: D. E. Pelinovsky

Copyright © 2012 Hongwei Yang et al. 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. W. X. Ma and B. Fuchssteiner, “Integrable theory of the perturbation equations,” Chaos, Solitons and Fractals, vol. 7, no. 8, pp. 1227–1250, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. W. X. Ma, “Integrable couplings of soliton equations by perturbations. I: a general theory and application to the KdV hierarchy,” Methods and Applications of Analysis, vol. 7, no. 1, pp. 21–55, 2000. View at Zentralblatt MATH
  3. G. Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems,” Journal of Mathematical Physics, vol. 30, no. 2, pp. 330–338, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. W. X. Ma, “A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction,” Chinese Journal of Contemporary Mathematics, vol. 13, no. 1, pp. 79–89, 1992. View at Zentralblatt MATH
  5. F. K. Guo, “Two hierarchies of integrable Hamiltonian equations,” Mathematica Applicata, vol. 9, no. 4, pp. 495–499, 1996. View at Zentralblatt MATH
  6. Y. F. Zhang, “Two types of new Lie algebras and corresponding hierarchies of evolution equations,” Physics Letters A, vol. 310, no. 1, pp. 19–24, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. F. K. Guo and Y. F. Zhang, “A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling,” Journal of Mathematical Physics, vol. 44, no. 12, pp. 5793–5803, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. W. X. Ma, “A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order,” Physics Letters A, vol. 367, no. 6, pp. 473–477, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. T. C. Xia, F. C. You, and W. Y. Zhao, “Multi-component Levi hierarchy and its multi-component integrable coupling system,” Communications in Theoretical Physics, vol. 44, no. 6, pp. 990–996, 2005. View at Publisher · View at Google Scholar
  10. H.-H. Dong and N. Zhang, “The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy,” Chinese Physics, vol. 15, no. 9, pp. 1919–1926, 2006. View at Publisher · View at Google Scholar
  11. T. Xia, F. Yu, and Y. Zhang, “The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system,” Physica A, vol. 343, no. 15, pp. 238–246, 2004.
  12. Z. Li and H. Dong, “Integrable couplings of the multi-component dirac hierarchy and its Hamiltonian structure,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 290–295, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. W. X. Ma and Y. Zhang, “Component-trace identities for Hamiltonian structures,” Applicable Analysis, vol. 89, no. 4, pp. 457–472, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. X. Ma, X. X. Xu, and Y. F. Zhang, “Semi-direct sums of Lie algebras and continuous integrable couplings,” Physics Letters A, vol. 351, no. 3, pp. 125–130, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. W. X. Ma, “Variational identities and Hamiltonian structures,” AIP Conference Proceedings, vol. 1212, no. 1, pp. 1–27, 2010.
  16. W. X. Ma, “Nonlinear continuous integrable Hamiltonian couplings,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7238–7244, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Y. F. Zhang, “Lie algebras for constructing nonlinear integrable couplings,” Communications in Theoretical Physics, vol. 56, no. 5, pp. 805–812, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Y. F. Zhang and B. L. Feng, “A few Lie algebras and their applications for generating integrable hierarchies of evolution types,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 8, pp. 3045–3061, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. W. X. Ma, “Loop algebras and bi-integrable couplings,” Chinese Annals of Mathematics B, vol. 33, no. 2, pp. 207–224, 2012. View at Publisher · View at Google Scholar
  20. M. Antonowicz, “Gelfand-Dikii hierarchies with the sources and Lax representation for restricted flows,” Physics Letters A, vol. 165, no. 1, pp. 47–52, 1992. View at Publisher · View at Google Scholar
  21. Y. B. Zeng, W. X. Ma, and R. Lin, “Integration of the soliton hierarchy with self-consistent sources,” Journal of Mathematical Physics, vol. 41, no. 8, pp. 5453–5489, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. V. K. Mel'nikov, “New method for deriving nonlinear integrable systems,” Journal of Mathematical Physics, vol. 31, no. 5, pp. 1106–1113, 1990. View at Publisher · View at Google Scholar
  23. C. Claude, A. Latifi, and J. Leon, “Nonlinear resonant scattering and plasma instability: an integrable model,” Journal of Mathematical Physics, vol. 32, no. 12, pp. 3310–3321, 1991. View at Zentralblatt MATH
  24. F. J. Yu, “A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with sl ˜(4),” Physics Letters A, vol. 372, no. 44, pp. 6613–6621, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. T. C. Xia, “Two new integrable couplings of the soliton hierarchies with self-consistent sources,” Chinese Physics B, vol. 19, no. 10, Article ID 100303, 2010. View at Publisher · View at Google Scholar
  26. F. J. Yu and L. Li, “Integrable coupling system of JM equations hierarchy with self-consistent sources,” Communications in Theoretical Physics, vol. 53, no. 1, pp. 6–12, 2010. View at Publisher · View at Google Scholar
  27. G. Z. Tu, “An extension of a theorem on gradients of conserved densities of integrable systems,” Northeastern Mathematical Journal, vol. 6, no. 1, pp. 26–32, 1990. View at Zentralblatt MATH