Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 153139, 9 pages
http://dx.doi.org/10.1155/2014/153139
Research Article

Bifurcation of Traveling Wave Solutions of the Dual Ito Equation

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 9 May 2014; Accepted 15 July 2014; Published 5 August 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Xinghua Fan and Shasha Li. 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. C. Guan and Z. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. O. G. Mustafa, “On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system,” Wave Motion, vol. 46, no. 6, pp. 397–402, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. Constantin and R. I. Ivanov, “On an integrable two-component Camassa-Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. B. Li and Y. S. Li, “Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation,” Acta Mathematica Sinica, vol. 24, no. 8, pp. 1319–1330, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. X. Fan, S. Yang, J. Yin, and L. Tian, “Bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 3956–3963, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. P. Guha and P. J. Olver, “Geodesic flow and two (super) component analog of the Camassa-Holm equation,” Symmetry Integrability and Geometry Methods and Applications, vol. 2, article 054, 2006. View at Publisher · View at Google Scholar
  8. M. Ito, “Symmetries and conservation laws of a coupled nonlinear wave equation,” Physics Letters A, vol. 91, no. 7, pp. 335–338, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. R. Camassa, D. D. Holm, and J. M. Hyman, “A new integrable shallow water equation,” Advances in Applied Mechanics, vol. 31, pp. 1–33, 1994. View at Google Scholar
  10. R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. C. A. Gomez S, “New traveling waves solutions to generalized Kaup-KUPershmidt and Ito equations,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 241–250, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. F. Khani, “Analytic study on the higher order Ito equations: new solitary wave solutions using the Exp-function method,” Chaos, Solitons & Fractals, vol. 41, no. 4, pp. 2128–2134, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. D. Li and J. Zhao, “New exact solutions to the (2+1)-dimensional Ito equation: extended homoclinic test technique,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1968–1974, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. S. F. Tian and H. Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the 1+1-dimensional and 2+1-dimensional Ito equation,” Chaos, Solitons & Fractals, vol. 47, pp. 27–41, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Z. Zhao, Z. Dai, and C. Wang, “Extend three-wave method for the (1+2)-dimensional Ito equation,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2295–2300, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. Y. Zhang, Y. C. You, W. X. Ma, and H. Q. Zhao, “Resonance of solitons in a coupled higher-order Ito equation,” Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 121–128, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. H. Zhao, “Soliton solution of a multi-component higher-order Ito equation,” Applied Mathematics Letters, vol. 26, no. 7, pp. 681–686, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. Chen, S. Wen, S. Tang, W. Huang, and Z. Qiao, “Effects of quadratic singular curves in integrable 5 equations,” To appear in Studies in Applied Mathematics.
  19. L. Perko, Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  20. G. Betchewe, B. B. Thomas, K. K. Victor, and K. T. Crepin, “Dynamical survey of a generalized-Zakharov equation and its exact travelling wave solutions,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 203–211, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. K. Gatermann and S. Hosten, “Computational algebra for bifurcation theory,” Journal of Symbolic Computation, vol. 40, no. 4-5, pp. 1180–1207, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. J. B. Li and H. H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach, Science Press, Beijing, China, 2007.
  23. Y. Feng, W. Shan, W. Sun, H. Zhong, and B. Tian, “Bifurcation analysis and solutions of a three-dimensional Kudryashov-Sinelshchikov equation in the bubbly liquid,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp. 880–886, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  24. H. Liu and J. Li, “Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations,” Journal of Computational and Applied Mathematics, vol. 257, pp. 144–156, 2014. View at Publisher · View at Google Scholar · View at MathSciNet