Table of Contents
International Scholarly Research Notices
Volume 2014 (2014), Article ID 840689, 8 pages
http://dx.doi.org/10.1155/2014/840689
Research Article

Exact Solutions for the Integrable Sixth-Order Drinfeld-Sokolov-Satsuma-Hirota System by the Analytical Methods

Department of Applied Mathematics, Faculty of Mathematics Science, University of Tabriz, 29 Bahman Boulevard, Tabriz 5166616471, Iran

Received 7 April 2014; Revised 28 May 2014; Accepted 28 May 2014; Published 9 September 2014

Academic Editor: Abdelghani Bellouquid

Copyright © 2014 Jalil Manafian Heris and Mehrdad Lakestani. 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. V. G. Drinfeld and V. V. Sokolov, “Equations of Korteweg-de Vries type and simple Lie algebras,” Doklady Akademii Nauk, vol. 258, pp. 11–16, 1981. View at Google Scholar
  2. A. Karsau-Kalkani and S. Yu Sarkovich, “Bäcklund transformation and special solutions for Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations,” Journal of Physics A: Mathematical and General, vol. 34, pp. 7353–7358, 2001. View at Google Scholar
  3. A. M. Wazwaz, “Exact and explicit travelling wave solutions for the nonlinear Drinfeld-Sokolov system,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 3, pp. 311–325, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. A. Karsau-Kalkani, A. Karsau, A. Sakovich, S. Sarkovich, and R. Turhan, “A new integrable generalization of the Korteweg-de Vries equation,” Journal of Mathematical Physics, vol. 49, pp. 1–10, 2008. View at Google Scholar
  5. J. Satsuma and R. Hirota, “A coupled KdV equation is one case of the four-reduction of the KP hierarchy,” Journal of the Physical Society of Japan, vol. 51, no. 10, pp. 3390–3397, 1982. View at Google Scholar · View at Scopus
  6. A. M. Wazwaz, “The Cole-Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota equation,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 248–255, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. R. Hirota and M. Ito, “Resonance of solitons in one dimension,” Journal of the Physical Society of Japan, vol. 52, no. 3, pp. 744–748, 1983. View at Google Scholar · View at Scopus
  8. M. Ito, “An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders,” Journal of the Physical Society of Japan, vol. 49, no. 2, pp. 771–778, 1980. View at Google Scholar · View at Scopus
  9. J. Hietarinta, “A search for bilinear equations passing Hirota's three-soliton condition. I. KdV-type bilinear equations,” Journal of Mathematical Physics, vol. 28, no. 8, pp. 1732–1742, 1987. View at Google Scholar · View at Scopus
  10. W. Hereman and W. Zhuang, “AMACSYMA program for the Hirota method,” in Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, pp. 22–26, 1991.
  11. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.
  12. A.-M. Wazwaz, “Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 755–760, 2006. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Dehghan and J. Manafian, “The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method,” Zeitschrift für Naturforschung, vol. 64, pp. 420–430, 2009. View at Google Scholar
  14. M. Dehghan, J. Manafian, and A. Saadatmandi, “The solution of the linear fractional partial differential equations using the homotopy analysis method,” Zeitschrift für Naturforschung, vol. 65, no. 11, pp. 935–949, 2010. View at Google Scholar · View at Scopus
  15. M. Dehghan, J. Manafian, and A. Saadatmandi, “Solving nonlinear fractional partial differential equations using the homotopy analysis method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448–479, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. J. H. He, “Variational iteration method—a kind of non-linear analytical technique: Some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at Google Scholar · View at Scopus
  17. M. Dehghan, J. M. Heris, and A. Saadatmandi, “Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 21, no. 6, pp. 736–753, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. X. H. Menga, W. J. Liua, H. W. Zhua, C. Y. Zhang, and B. Tian, “Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation,” Journal of Physics A, vol. 387, no. 97, 107 pages, 2008. View at Google Scholar
  19. X. Lü, H. W. Zhu, X. H. Meng, Z. C. Yang, and B. Tian, “Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1305–1315, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. J. Manafian Heris and M. Bagheri, “Exact solutions for the modified KdV and the generalized KdV equations via Exp-function method,” Journal of Mathematical Extension, vol. 4, pp. 77–98, 2010. View at Google Scholar
  22. M. Fazli Aghdaei and J. Manafianheris, “Exact solutions of the couple Boiti-Leon-Pempinelli system by the generalized (G'/G)-expansion method,” Journal of Mathematical Extension, vol. 5, pp. 91–104, 2011. View at Google Scholar
  23. A. Bekir, “Application of the (G'/G)-expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. J. Manafianheris, “Solving the integro-differential equations using the modified Laplace Adomian decomposition method,” Journal of Mathematical Extension, vol. 6, pp. 41–55, 2012. View at Google Scholar
  25. M. Bagheri and J. Manafian Heris, “Differential transform method for solving the linear and nonlinear Westervelt equation,” Journal of Mathematical Extension, vol. 6, pp. 81–91, 2012. View at Google Scholar
  26. M. Wang, X. Li, and J. Zhang, “The (G'/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. J. Zhang, X. Wei, and Y. Lu, “A generalized (G'/G)-expansion method and its applications,” Physics Letters A, vol. 372, no. 20, pp. 3653–3658, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Zhang, J. L. Tong, and W. Wang, “A generalized (G'/G)-expansion method for the mKdV equation with variable coefficients,” Physics Letters A, vol. 372, no. 13, pp. 2254–2257, 2008. View at Publisher · View at Google Scholar · View at Scopus
  29. W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” The American Journal of Physics, vol. 60, pp. 650–654, 1992. View at Google Scholar
  30. S. D. Zhu, “The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equation,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1335–1342, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. Li and X. Zhang, “New exact kink solutions and periodic form solutions for a generalized Zakharov-Kuznetsov equation with variable coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3418–3422, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. M. E. Zayed Elsayed and H. ARNOUS Ahmed, “Many families of exact solutions for nonlinear system of partial differential equations describing the dynamics of DNA,” Journal of Partial Differential Equations, vol. 26, pp. 373–384, 2013. View at Google Scholar
  33. M. E. Zayed Elsayed, “Equivalence of the (G'/G)-expansion method and the Tanh-Coth function method,” in Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM '10), pp. 2225–2228, September 2010. View at Publisher · View at Google Scholar · View at Scopus