Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2015, Article ID 154915, 8 pages
http://dx.doi.org/10.1155/2015/154915
Research Article

Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

1School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China
2College of Sciences, China University of Mining and Technology, Xuzhou 221116, China

Received 28 October 2014; Revised 25 November 2014; Accepted 17 December 2014

Academic Editor: Stephen C. Anco

Copyright © 2015 Binlu Feng and Yufeng Zhang. 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. B. Fuchssteiner, “Coupling of completely integrable systems: the perturbation bundle,” in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, P. A. Clarkson, Ed., pp. 125–138, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at Google Scholar · View at MathSciNet
  2. W.-X. Ma, “Integrable couplings of soliton equations by perturbation I. A general theory and application to the KdV equation,” Methods and Applications of Analysis, vol. 7, no. 1, pp. 21–56, 2000. View at Google Scholar · View at MathSciNet
  3. Y. F. Zhang and H. Q. Zhang, “A direct method for integrable couplings of TD hierarchy,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 466–472, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. F. Guo and Y. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems,” Journal of Physics A: Mathematical and General, vol. 38, no. 40, pp. 8537–8548, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. W.-X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,” Journal of Physics A: Mathematical and General, vol. 39, no. 34, pp. 10787–10801, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. W. X. Ma and B. Fuchssteiner, “The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy,” Physics Letters A, vol. 213, no. 1-2, pp. 49–55, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  7. W.-X. Ma, X.-X. Xu, and Y. 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 MathSciNet · View at Scopus
  8. W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings,” Journal of Mathematical Physics, vol. 47, no. 5, Article ID 053501, 16 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. W.-X. Ma, “Variational identities and applications to Hamiltonian structures of soliton equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1716–e1726, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. Y. Zhang and H. Tam, “A few integrable systems and spatial spectral transformations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 3770–3783, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y.-F. Zhang and Y. C. Hon, “Some evolution hierarchies derived from self-dual Yang-Mills equations,” Communications in Theoretical Physics, vol. 56, no. 5, pp. 856–872, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. B. L. Feng and J. Q. Liu, “Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 661–672, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z. J. Qiao, “Commutator representations of three isospectral equation hierarchies,” Chinese Journal of Contemporary Mathematics, vol. 14, no. 1, pp. 41–49, 1993. View at Google Scholar · View at MathSciNet
  14. Z. J. Qiao, “A general approach for getting the commutator representations of the hierarchies of nonlinear evolution equations,” Physics Letters A, vol. 195, no. 5-6, pp. 319–328, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  15. Z. Qiao, C. Cao, and W. Strampp, “Category of nonlinear evolution equations, algebraic structure, and r-matrix,” Journal of Mathematical Physics, vol. 44, no. 2, pp. 701–722, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Z. J. Qiao, “Generation of the hierarchies of solitons and generalized structure of the commutator representation, preprint 1992,” Acta Applied Mathematics Sinica, vol. 18, pp. 287–301, 1995. View at Google Scholar
  17. Z. J. Qiao and J. B. Li, “Negative-order KdV equation with both solitons and kink wave solutions,” Europhysics Letters, vol. 94, no. 5, Article ID 50003, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. Z. J. Qiao and E. G. Fan, “Negative-order Korteweg-de Vries equations,” Physical Review E, vol. 86, no. 1, Article ID 016601, 2012. View at Publisher · View at Google Scholar · View at Scopus
  19. W.-X. Ma and Z.-N. Zhu, “Constructing nonlinear discrete integrable Hamiltonian couplings,” Computers & Mathematics with Applications, vol. 60, no. 9, pp. 2601–2608, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. Y. F. Zhang and H. W. Tam, “Three kinds of coupling integrable couplings of the Korteweg-de Vries hierarchy of evolution equations,” Journal of Mathematical Physics, vol. 51, no. 4, Article ID 043510, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. Y. F. Zhang and E. G. Fan, “Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti-Pempinelli-Tu hierarchy,” Journal of Mathematical Physics, vol. 51, Article ID 083506, pp. 1–18, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Y. F. Zhang and H. W. Tam, “Four Lie algebras associated with R6 and their applications,” Journal of Mathematical Physics, vol. 51, no. 9, Article ID 093514, 30 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  23. 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 MathSciNet · View at Scopus
  24. 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 Google Scholar · View at MathSciNet
  25. Y.-F. Zhang and J. Liu, “Induced LIE algebras of a six-dimensional matrix LIE algebra,” Communications in Theoretical Physics, vol. 50, no. 2, pp. 289–294, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Y. F. Zhang and J. Q. Mei, “Lie algebras and integrable systems,” Communications in Theoretical Physics, vol. 57, no. 6, pp. 1012–1022, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. B. Feng, Y. Zhang, and H. Dong, “A few integrable couplings of some integrable systems and (2+1)-dimensional integrable hierarchies,” Abstract and Applied Analysis, vol. 2014, Article ID 932672, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet