About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 153974, 8 pages
http://dx.doi.org/10.1155/2014/153974
Research Article

Exp-Function Method for a Generalized MKdV Equation

School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Received 15 December 2013; Accepted 24 April 2014; Published 15 May 2014

Academic Editor: Cengiz Çinar

Copyright © 2014 Yuzhen Chai et al. 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. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. Scott, Encyclopedia of Nonlinear Science, Routledge, New York, NY, USA, 2005. View at MathSciNet
  3. S. J. Li, Nonlinear Science and Its Application, Harbin Institute of Technology Press, Harbin, China, 2011.
  4. L. S. Shun, Forefront Issues of Nonlinear Science, China Science and Technology Press, Beijing, China, 2009.
  5. S. K. Liu and S. D. Liu, Nonlinear Equations in Physics, Peking University Press, Beijing, China, 2000.
  6. S. K. Liu, S. D. Liu, and B. D. Tan, Nonlinear Atmospherics Dynamics, China National Defense Industry Press, Bejing, China, 1996.
  7. G. Chen, Some Issues on the Theroy of Nonlinear Science and Its Application. [Academic Dissertation], Zhejiang University Press, Zhejiang, China, 1996.
  8. J. T. Huang, J. Z. Xu, and Y. T. Xiong, Solitons Concepts, Principles and Applications, Higher Education Press, Beijing, China, 2004.
  9. J. S. Russell, “Report on waves,” in Proceedings of the 14th Meeting of the Bntish Association for the Advancement of Science, pp. 311–390, John Murray, London, UK, 1844.
  10. D. J. Kortewrg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave,” Philosophical Magazine, vol. 39, no. 240, pp. 422–443, 1895. View at Publisher · View at Google Scholar
  11. E. Fermi, J. Pasta, and S. Ulam, “Studies of nonlinear problems,” Los Alamos Document, Los Alamos, NM, USA, 1955.
  12. N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states,” Physical Review Letters, vol. 15, no. 6, pp. 240–243, 1965. View at Publisher · View at Google Scholar · View at Scopus
  13. T. X. Chen, Introduction to Nonlinear Physics, University of Science and Technology of China Press, Hefei, China, 2002.
  14. Z. L. Pan, The Mathematic Methods of the Nonlinear Problem and Its Application, Zhejiang University Press, Zhejiang, China, 2002.
  15. D. Y. Wang, D. J. Wu, and G. L. Huang, Solitary Wave in Space Plasma, Shanghai Scientific and Technological Education Publishing House, Hangzhou, China, 2000.
  16. H. A. Abdusalam, “On an improved complex tanh-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 99–106, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. A. Abdou and A. A. Soliman, “Modified extended tanh-function method and its application on nonlinear physical equations,” Physics Letters A: General, Atomic and Solid State Physics, vol. 353, no. 6, pp. 487–492, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. D. L. Sekulić, M. V. Satarić, and M. B. Živanov, “Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3499–3506, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. W.-W. Li, Y. Tian, and Z. Zhang, “F method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations,” Applied Mathematics and Computation, vol. 219, no. 3, pp. 1135–1143, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. H. Bhrawy, M. A. Abdelkawy, and A. Biswas, “Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi's elliptic function method,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 4, pp. 915–925, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Guo and H.-Q. Hao, “Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 9, pp. 2426–2435, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. F. Qi, B. Tian, X. Lü, R. Guo, and Y. Xue, “Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2372–2381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. R. Guo and B. Tian, “Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 8, pp. 3189–3203, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. R. Guo, B. Tian, L. Wang, F.-H. Qi, and Y. Zhan, “Darboux transformation and soliton solutions for a system describing ultrashort pulse propagation in a multicomponent nonlinear medium,” Physica Scripta, vol. 81, no. 2, Article ID 025002, 2010. View at Publisher · View at Google Scholar
  25. R. Guo, B. Tian, L. Xing, H.-Q. Zhang, and T. Xu, “Integrability aspects and soliton solutions for a system describing ultrashort pulse propagation in an inhomogeneous multi-component medium,” Communications in Theoretical Physics, vol. 54, no. 3, pp. 536–544, 2010.
  26. R. Guo, B. Tian, X. Lü, H.-Q. Zhang, and W.-J. Liu, “Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrodinger-Maxwell-Bloch system with symbolic computation,” Computational Mathematics and Mathematical Physics, vol. 52, no. 4, pp. 565–577, 2012.
  27. L. Bougoffa and R. C. Rach, “Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method,” Applied Mathematics and Computation, vol. 225, pp. 50–61, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. T. S. El-Danaf, M. A. Ramadan, and F. E. I. Abd Alaal, “The use of adomian decomposition method for solving the regularized long-wave equation,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 747–757, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. S. M. El-Sayed, D. Kaya, and S. Zarea, “The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 2, pp. 105–112, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. H. Liu, “Variational Approach to Nonlinear Electrochemical System,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 1, pp. 95–96, 2004. View at Scopus
  31. J. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207–208, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. J. He and X. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564–567, 1993. View at Publisher · View at Google Scholar · View at Scopus