Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 597983, 14 pages
http://dx.doi.org/10.1155/2012/597983
Research Article

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

Mathematics Department, Faculty of Science, Taif University, Saudi Arabia

Received 2 April 2012; Revised 15 July 2012; Accepted 31 July 2012

Academic Editor: Renat Zhdanov

Copyright © 2012 J. F. Alzaidy. 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. M. J. Ablowitz and P. A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. C. H. Gu, H. S. Hu, and Z. X. Zhou, Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang, China, 1990.
  3. V. B. Matveev and M. A. Salle, Darboux transformations and solitons, Springer-Verlag, Berlin, Germany, 1991. View at Zentralblatt MATH
  4. R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S.-Y. Lou and J. Z. Lu, “Special solutions from the variable separation approach: the Davey-Stewartson equation,” Journal of Physics A, vol. 29, no. 14, pp. 4209–4215, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. E. J. Parkes and B. R. Duffy, “Travelling solitary wave solutions to a compound KdV-Burgers equation,” Physics Letters A, vol. 229, no. 4, pp. 217–220, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters. A, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Z. Y. Yan, “New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations,” Physics Letters A, vol. 292, no. 1-2, pp. 100–106, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. Chen and Z. Yu, “Generalized extended tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves,” International Journal of Modern Physics C, vol. 14, no. 5, pp. 601–611, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996. View at Google Scholar
  11. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, NY, USA, 1989.
  12. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, NY, USA, 1986. View at Publisher · View at Google Scholar
  13. E. M. E. Zayed and K. A. Gepreel, “The (G/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,” Journal of Mathematical Physics, vol. 50, no. 1, p. 12, 2009. View at Publisher · View at Google Scholar
  14. Z. Y. Yan, “A reduction mKdV method with symbolic computation to construct new doubly-periodic solutions for nonlinear wave equations,” International Journal of Modern Physics C, vol. 14, no. 5, pp. 661–672, 2003. View at Publisher · View at Google Scholar · View at Scopus
  15. Z. Y. Yan, “The new tri-function method to multiple exact solutions of nonlinear wave equations,” Physica Scripta, vol. 78, no. 3, Article ID 035001, p. 5, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Z. Y. Yan, “Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas,” Physics Letters A, vol. 373, no. 29, pp. 2432–2437, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. D. C. Lu and B. J. Hong, “New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 572–580, 2008. View at Publisher · View at Google Scholar
  18. A. V. Porubov, “Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer,” Physics Letters A, vol. 221, no. 6, pp. 391–394, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A.-M. Wazwaz, “The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1179–1195, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z. Y. Yan and H. Q. Zhang, “New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water,” Physics Letters A, vol. 285, no. 5-6, pp. 355–362, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. D. Lü, “Jacobi elliptic function solutions for two variant Boussinesq equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1373–1385, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Z. Y. Yan, “Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 299–309, 2003. View at Publisher · View at Google Scholar
  23. C. L. Bai and H. Zhao, “Generalized method to construct the solitonic solutions to (3+1)-dimensional nonlinear equation,” Physics Letters A, vol. 354, no. 5-6, pp. 428–436, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. F. Cariello and M. Tabor, “Similarity reductions from extended Painlevé expansions for nonintegrable evolution equations,” Physica D, vol. 53, no. 1, pp. 59–70, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. M. Wang and X. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations,” Physics Letters A, vol. 343, no. 1–3, pp. 48–54, 2005. View at Publisher · View at Google Scholar
  26. X. Feng, “Exploratory approach to explicit solution of nonlinear evolution equations,” International Journal of Theoretical Physics, vol. 39, no. 1, pp. 207–222, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. L. Hu, “Explicit solutions to three nonlinear physical models,” Physics Letters A, vol. 287, no. 1-2, pp. 81–89, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. J. L. Hu and H. Zhang, “A new method for finding exact traveling wave solutions to nonlinear partial differential equations,” Physics Letters A, vol. 286, no. 2-3, pp. 175–179, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. 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 Zentralblatt MATH
  30. X. Li and M. L. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms,” Physics Letters A, vol. 361, no. 1-2, pp. 115–118, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. K. A. Gepreel, “Exact solutions for nonlinear PDEs with the variable coefficients in mathematical physics,” Journal of Computational Science, vol. 6, no. 1, pp. 003–014, 2011. View at Google Scholar
  32. J. P. Wang, “A list of 1+1 dimensional integrable equations and their properties,” Journal of Nonlinear Mathematical Physics, vol. 9, supplement 1, p. 213, 2002. View at Publisher · View at Google Scholar