1887
Volume 2013, Issue 1
  • EISSN: 2223-506X

Abstract

The (′/, 1/)-expansion method is one of the most direct and effective methods for obtaining exact traveling wave solutions of nonlinear partial differential equations (PDEs). In this paper, we construct exact travelling wave solutions of nonlinear evolution equations in mathematical physics via generalized Zakharov-Kuznetsov (ZK), modified Zakharov-Kuznetsov and Sharma–Tasso–Olver (STO) equations by (′/, 1/)-expansion method, where satisfies the auxiliary ordinary differential equation (ODE) ; λ and μ are arbitrary constants.

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2013-11-01
2024-10-15
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References

  1. Abdou MA. The extended tanh method and its applications for solving nonlinear physical models. Appl Math Comput. 2007; 190:1:988996.
    [Google Scholar]
  2. Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge: Cambridge University Press 1991.
    [Google Scholar]
  3. Ali Akbar M, Ali NHM, Mohyud-Din ST. The alternative-expansion method with generalized Riccati equation: Application to fifth order (1+1)-dimensional Caudrey-Dodd-Gibbon equation. Int J Phys Sci. 2012; 7:5:743752.
    [Google Scholar]
  4. Naher H, Abdullah FA, Akbar MA. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. J Appl Math. 2012; 2012::114.
    [Google Scholar]
  5. Akbar MA, Ali NHM, Zayed EME. Abundant exact traveling wave solutions of the generalized Bretherton equation via the improved G′/G-expansion method. Commun Theor Phys. 2012; 57:2:173178.
    [Google Scholar]
  6. Chen Y, Wang Q. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons & Fractals. 2005; 24:3:745757.
    [Google Scholar]
  7. El-Wakil SA, Abdou MA, El-Shewy EK, Hendi A. (G′/G)-expansion method equivalent to the extended tanh-function method. Physica Scripta. 2010; 81:3:035011.
    [Google Scholar]
  8. Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A. 2000; 277:4-5:212218.
    [Google Scholar]
  9. He J-H, Wu X-H. Exp-function method for nonlinear wave equations. Chaos, Soliton Fract. 2006; 30:3:700708.
    [Google Scholar]
  10. Hirota R. Exact solution of the korteweg—de vries equation for multiple collisions of solitons. Phys Rev Lett. 1971; 27:18:11921194.
    [Google Scholar]
  11. Darvishi MT, Najaf M. Some exact solutions of the (2+1)-dimensional breaking soliton equation using the three- wave method. Int J Comput Math Sci. 2012; 6:1:1316.
    [Google Scholar]
  12. Kawahara T. Oscillatory solitary waves in dispersive media. J Phys Soc Japan. 1972; 33:1:260264.
    [Google Scholar]
  13. Kudryashov NA. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys Lett A. 1990; 147:5-6:287291.
    [Google Scholar]
  14. Kudryashov NA. On types of nonlinear nonintegrable equations with exact solutions. Phys Lett A. 1991; 155:4-5:269275.
    [Google Scholar]
  15. Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul. 2009; 14:9-10:35073529.
    [Google Scholar]
  16. Liu S, Fu Z, Liu S, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A. 2001; 289:1-2:6974.
    [Google Scholar]
  17. Lu D, Hong B, Tian L. New solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers equations. Commun Nonlinear Sci Numer Simul. 2009; 14:1:7784.
    [Google Scholar]
  18.  D. Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos, Solitons Fract. 2005; 24:5:13731385.
    [Google Scholar]
  19. Miura MR. Backlund Transformation. Berlin: Springer-Verlag 1978.
    [Google Scholar]
  20. Parkes EJ. Observations on the basic -expansion method for finding solutions to nonlinear evolution equations. Appl Math Comput. 2010; 217:4:17591763.
    [Google Scholar]
  21. Rogers C, Shadwick WF. Backlund Transformations. New York: Academic Press 1982.
    [Google Scholar]
  22. Bekir A. Exact solutions for some (2+1)-dimensional nonlinear evolution equations by using tanh-coth method. World Applied Sciences Journal; Special Issue of Applied Mathematics. 2010; 9:0106.
    [Google Scholar]
  23. Zhang X-L, Zhang H-Q. A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order. Appl Math Comput. 2007; 186:1:705714.
    [Google Scholar]
  24. Wang M, Zhou Y. The periodic wave solutions for the Klein-Gordon-Schrödinger equations. Phys Lett A. 2003; 318:1-2:8492.
    [Google Scholar]
  25. Wang M, Li X. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys Lett A. 2005; 343:1-3:4854.
    [Google Scholar]
  26. Wang M, Li X. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons Fract. 2005; 24:5:12571268.
    [Google Scholar]
  27. Wang M, Li X, Zhang J. Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation. Phys Lett A. 2007; 363:1-2:96101.
    [Google Scholar]
  28. Wang DS, Ren YJ, Zhang HQ. Further extended sinh-cosh and Sin-Cos methods and new non-traveling wave solutions of the (2+1)-dimensional dispersive long wave equations. Appl Math E-Notes. 2005; 5::157163.
    [Google Scholar]
  29. Ma S-H, Fang J-P, Zheng C-L. New exact solutions of the (2+1)-dimensional breaking soliton system via an extended mapping method. Chaos, Solitons Fract. 2009; 40:1:210214.
    [Google Scholar]
  30. Wang M, Li X, Zhang J. The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A. 2008; 372:4:417423.
    [Google Scholar]
  31. Wazwaz A-M. New solutions of distinct physical structures to high-dimensional nonlinear evolution equations. Appl Math Comput. 2008; 196:1:363370.
    [Google Scholar]
  32. Wazwaz A-M. The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations. Chaos, Solitons Fract. 2008; 38:5:15051516.
    [Google Scholar]
  33. Inan IE. Generalized jacobi elliptic function method for traveling wave solutions of (2+1)-dimensional breaking soliton equation. Cankaya Univ J Sci Eng. 2010; 7:1:3950.
    [Google Scholar]
  34. Wazzan L. A modified tanh-coth method for solving the KdV and the KdV-Burgers' equations. Commun Nonlinear Sci Numer Simul. 2009; 14:2:443450.
    [Google Scholar]
  35. Yomba E. A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations. Phys Lett A. 2008; 372:7:10481060.
    [Google Scholar]
  36. Naher H, Abdullah FA, Akbar MA. The exp-function method for new exact solutions of the nonlinear partial differential equations. Int J Phys Sci. 2011; 6:29:67066716.
    [Google Scholar]
  37. Yusufoğlu E, Bekir A. Exact solutions of coupled nonlinear evolution equations. Chaos, Solitons Fract. 2008; 37:3:842848.
    [Google Scholar]
  38. Zayed EME, Zedan HA, Gepreel KA. On the solitary wave solutions for nonlinear Euler equations. Appl Anal. 2004; 83:11:11011132.
    [Google Scholar]
  39. Zayed EME, Zedan HA, Gepreel KA. Group Analysis and Modified Extended Tanh-function to Find the Invariant Solutions and Soliton Solutions for Nonlinear Euler Equations. Int J Nonlinear Sci Numer Simul. 2004; 5:3.
    [Google Scholar]
  40. Zayed EME, Abourabia AM, Gepreel KA, El Horbaty MM. Travelling solitary wave solutions for the nonlinear coupled Korteweg-de Vries system. Chaos, Solitons Fract. 2007; 34:2:292306.
    [Google Scholar]
  41. Akbar MA, Ali NHM. The alternative (G′/G)-expansion method and its applications to nonlinear partial differential equations. Int J Phys Sci. 2011; 6:35:79107920.
    [Google Scholar]
  42. Akbar MA, Ali NHM, Zayed EME. A generalized and improved (G′/G)-expansion method for nonlinear evolution equations. Math Probl Eng. 2012; 2012::122.
    [Google Scholar]
  43. Zhang S-L, Wu B, Lou S-Y. Painlevé analysis and special solutions of generalized Broer-Kaup equations. Phys Lett A. 2002; 300:1:4048.
    [Google Scholar]
  44. Akbar MA, Ali NHM. Exp-function method for Duffing equation and new solutions of (2+1) dimensional dispersive long wave equations. Prog Appl Math. 2011; 1:2:3042.
    [Google Scholar]
  45. Akbar MA, Ali NHM. New solitary and periodic solutions of nonlinear evolution equation by exp-function method. World Appl Sci J. 2012; 17:12:16031610.
    [Google Scholar]
  46. Zhang S, Xia TC. A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations. Appl Math E-Notes. 2008; 8::5866.
    [Google Scholar]
  47. Zhang S, Xia T-C. A generalized new auxiliary equation method and its applications to nonlinear partial differential equations. Phys Lett A. 2007; 363:5-6:356360.
    [Google Scholar]
  48. Zhang S, Tong J-L, Wang W. A generalized (G′/G)-expansion method for the mKdV equation with variable coefficients. Phys Lett A. 2008; 372:13:22542257.
    [Google Scholar]
  49. Zhang J, Wei X, Lu Y. A generalized (G′/G)-expansion method and its applications. Phys Lett A. 2008; 372:20:36533658.
    [Google Scholar]
  50. Li L-xiao, Li E-qiang, Wang M-liang. The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations. Appl Math-A J Chinese Univ. 2010; 25:4:454462.
    [Google Scholar]
  51. Zayed EME, Hoda Ibrahim SA, Abdelaziz MAM. Traveling wave solutions of the nonlinear (3+1)-dimensional Kadomtsev-Petviashvili equation using the two variables (G′/G, 1/G)-expansion method. J Appl Math. 2012; 2012::18.
    [Google Scholar]
  52. Zayed EME, Abdelaziz MAM. The two-variable (G′/G, 1/G)-expansion method for solving the nonlinear KdV-mKdV equation. Math Probl Eng. 2012; 2012::114.
    [Google Scholar]
  53. Zayed EME, Hoda Ibrahim SA. The two variable (G′/G, 1/G)-expansion method for finding exact travelling wave solutions of the (3+1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation. International Conference on Information, Business and Education Technology. (ICIBET 2012). Beijing: Atlantis Press 2013.
    [Google Scholar]
  54. Ling-Xiao L, Ming-Liang W. The (G′/G)-expansion method and travelling wave solutions for a higher-order nonlinear schrödinger equation. Appl Math Comput. 2009; 208:2:440445.
    [Google Scholar]
  55. Zhang JI, Wang YM, Wang ML. Exact solutions to the two nonlinear equations. Acta Physica Sinica. 2003; 52:7:15741578.
    [Google Scholar]
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