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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 681383, 6 pages
http://dx.doi.org/10.1155/2013/681383
Research Article

Further Results on the Traveling Wave Solutions for an Integrable Equation

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 19 December 2012; Accepted 8 February 2013

Academic Editor: Jong Hae Kim

Copyright © 2013 Chaohong Pan and Zhengrong Liu. 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. Z. J. Qiao, “New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons,” Journal of Mathematical Physics, vol. 48, no. 8, pp. 082701–082720, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Z. J. Qiao and L. P. Liu, “A new integrable equation with no smooth solitons,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 587–593, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Sakovich, “Smooth soliton solutions of a new integrable equation by Qiao,” Journal of Mathematical Physics, vol. 52, no. 2, p. 023509, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  4. P. G. Estévez, “Generalized Qiao hierarchy in 2+1 dimensions: reciprocal transformations, spectral problem and non-isospectrality,” Physics Letters A, vol. 375, no. 3, pp. 537–540, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y.-Q. Yang and Y. Chen, “Prolongation structure of the equation studied by Qiao,” Communications in Theoretical Physics, vol. 56, no. 3, pp. 463–466, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. Qiao, “A new integrable equation with cuspons and W/M-shape-peaks solitons,” Journal of Mathematical Physics, vol. 47, no. 11, pp. 112701–112709, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Li and Z. Qiao, “Bifurcations of traveling wave solutions for an integrable equation,” Journal of Mathematical Physics, vol. 51, no. 4, pp. 042703–042723, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. B. Li and H. H. Dai, On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach, Science Press, Beijing, China, 2007.
  9. J. B. Li and G. Chen, “On a class of singular nonlinear traveling wave equations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 17, no. 11, pp. 4049–4065, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Li, J. Wu, and H. Zhu, “Traveling waves for an integrable higher order KdV type wave equations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 8, pp. 2235–2260, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. S. Alber, R. Camassa, Y. N. Fedorov, D. D. Holm, and J. E. Marsden, “The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type,” Communications in Mathematical Physics, vol. 221, no. 1, pp. 197–227, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Z. J. Qiao, “The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold,” Communications in Mathematical Physics, vol. 239, no. 1-2, pp. 309–341, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  13. M. Kruskal, “Nonlinear wave equations dynamical systems, theory and applications,” Lecture Notes in Physics, vol. 38, pp. 310–354, 1975. View at Google Scholar
  14. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1981. View at MathSciNet
  15. R. M. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Review, vol. 18, no. 3, pp. 412–459, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, UK, 1982. View at MathSciNet
  17. J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, “Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation,” Philosophical Transactions of the Royal Society of London A, vol. 351, no. 1695, pp. 107–164, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philosophical Transactions of the Royal Society of London A, vol. 289, no. 1361, pp. 373–404, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Dey, “Domain wall solutions of KdV-like equations with higher order nonlinearity,” Journal of Physics A, vol. 19, no. 1, pp. L9–L12, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Dey, “KdV like equations with domain wall solutions and their Hamiltonians,” in Solitons, Nonlinear Dynam., pp. 188–194, Springer, Berlin, germany, 1988. View at Google Scholar · View at MathSciNet
  21. Z. Liu and C. Yang, “The application of bifurcation method to a higher-order KdV equation,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 1–12, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. Tang, R. Wang, and Z. Jing, “Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity,” Science in China A, vol. 45, no. 10, pp. 1255–1267, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z. Liu and T. Qian, “Peakons and their bifurcation in a generalized Camassa-Holm equation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 3, pp. 781–792, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Z. Liu, A. M. Kayed, and C. Chen, “Periodic waves and their limits for the Camassa-Holm equation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 8, pp. 2261–2274, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z. R. Liu and Y. Long, “Compacton-like wave and kink-like wave of GCH equation,” Nonlinear Analysis. Real World Applications, vol. 8, no. 1, pp. 136–155, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. Z. Liu and Z. Ouyang, “A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations,” Physics Letters A, vol. 366, no. 4-5, pp. 377–381, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. M. Song and Z. Liu, “Periodic wave solutions and their limits for the generalized KP-BBM equation,” Journal of Applied Mathematics, vol. 2012, Article ID 363879, 25 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Z. R. Liu, T. Jiang, P. Qin, and Q. Xu, “Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 810217, 23 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet