Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2018, Article ID 8583418, 8 pages
https://doi.org/10.1155/2018/8583418
Research Article

Traveling Wave Solutions of Two Nonlinear Wave Equations by -Expansion Method

1School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, China
2Graduate Department, Wuhan Textile University, Wuhan 430073, China

Correspondence should be addressed to Ben-gong Zhang; moc.621@9121nayneb

Received 4 October 2017; Revised 16 January 2018; Accepted 21 January 2018; Published 22 February 2018

Academic Editor: Zhi-Yuan Sun

Copyright © 2018 Yazhou Shi 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. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  2. O. Pashaev and G. Tanoǧlu, “Vector shock soliton and the Hirota bilinear method,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 95–105, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. C. H. Gu, Darboux Transformation in soliton Theory and its Gemetric Application, Shanghai scientific and Technical Publishers, Shanghai, China, 1999.
  4. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Gemany, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Coely, Baklund and Darboux Transform, American Mathematical Society, Providence, RI, USA, 2001.
  6. A.-M. Wazwaz, “The Camassa-Holm-KP equations with compact and noncompact travelling wave solutions,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 347–360, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  7. 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 Google Scholar · View at MathSciNet · View at Scopus
  8. 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 MathSciNet · View at Scopus
  9. J. H. He, “Exp-function method for nonlinear wave equations, Chaos,” Solitons & Fractals, vol. 30, no. 2, pp. 506–511, 2006. View at Google Scholar
  10. J. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1421–1429, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. H. He and X. H. Wu, “Exp-function method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903–910, 2008. View at Google Scholar
  12. A.-M. Wazwaz, “The sine-cosine method for obtaining solutions with compact and noncompact structures,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 559–576, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. A.-M. Wazwaz, “A sine-cosine method for handling nonlinear wave equations,” Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 499–508, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. A. Wazwaz and M. A. Helal, “Nonlinear variants of the BBM equation with compact and noncompact physical structures,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 767–776, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Y. Zhou, M. Wang, and T. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Physics Letters A, vol. 323, no. 1-2, pp. 77–88, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. Wang and X. Li, “Applications of -expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons & Fractals, vol. 24, no. 5, pp. 1257–1268, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  18. B.-g. Zhang, Z.-r. Liu, and J.-f. Mao, “New exact solutions for mCH and mDP equations by auxiliary equation method,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1306–1314, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  19. B.-G. Zhang, W. Li, and X. Li, “Peakons and new exact solitary wave solutions of extended quantum Zakharov-Kuznetsov equation,” Physics of Plasmas, vol. 24, no. 6, Article ID 062113, 2017. View at Publisher · View at Google Scholar · View at Scopus
  20. Z. Wen, “Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin–Gottwald–Holm system,” Nonlinear Dynamics, vol. 82, no. 1-2, pp. 767–781, 2015. View at Publisher · View at Google Scholar · View at Scopus
  21. Z. Wen, “Bifurcations and exact traveling wave solutions of a new two-component system,” Nonlinear Dynamics, vol. 87, no. 3, pp. 1917–1922, 2017. View at Publisher · View at Google Scholar · View at Scopus
  22. S. Li, Y. Li, and B.-G. Zhang, “Some singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation,” Nonlinear Dynamics, vol. 85, no. 3, pp. 1665–1677, 2016. View at Publisher · View at Google Scholar · View at Scopus
  23. B.-g. Zhang, S.-y. Li, and Z.-r. Liu, “Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations,” Physics Letters A, vol. 372, no. 11, pp. 1867–1872, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters A, vol. 199, no. 3-4, pp. 169–172, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. 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 Publisher · View at Google Scholar
  26. M. L. Wang, X. Z. Li, and J. L. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Phys Lett A, pp. 417–423, 2008. View at Google Scholar
  27. M. Wang, J. Zhang, and X. Li, “Application of the -expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 321–326, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. B.-G. Zhang, “Analytical and multishaped solitary wave solutions for extended reduced ostrovsky equation,” Abstract and Applied Analysis, vol. 2013, Article ID 670847, 2013. View at Publisher · View at Google Scholar · View at Scopus
  29. Y.-B. Zhou and C. Li, “Application of modified -expansion method to traveling wave solutions for Whitham-Broer-Kaup-like equations,” Communications in Theoretical Physics, vol. 51, no. 4, pp. 664–670, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  30. M. Mirzazadeh, M. Eslami, and A. Biswas, “Soliton solutions of the generalized Klein-Gordon equation by using ()-expansion method,” Computational & Applied Mathematics, vol. 33, no. 3, pp. 831–839, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  31. R. Fetecau and D. Levy, “Approximate model equations for water waves,” Communications in Mathematical Sciences, vol. 3, no. 2, pp. 159–170, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  32. V. G. Drinfel’d and V. V. Sokolov, “Equations of Korteweg-de Vries type, and simple Lie algebras,” Doklady Akademii Nauk SSSR, vol. 258, no. 1, pp. 11–16, 1981. View at Google Scholar · View at MathSciNet
  33. G. Wilson, “The affine Lie algebra C2(1) and an equation of Hirota and Satsuma,” Physics Letters A, vol. 89, no. 7, pp. 332–334, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  34. R. Hirota, B. Grammaticos, and A. Ramani, “Soliton structure of the Drinfel’d-Sokolov-Wilson equation,” Journal of Mathematical Physics, vol. 27, no. 6, pp. 1499–1505, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  35. M. Inc, “On numerical doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equation by the decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 421–430, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  36. H. T. Chen, “New Double Periodic Solutions of the Classical Drinfeld-Sokolov-Wilson Equation,” Numerical Analysis and Applied Mathematics, vol. 1048, pp. 138–142, 2008. View at Google Scholar
  37. C. Liu and X. Liu, “Exact solutions of the classical Drinfel’d-Sokolov-Wilson equations and the relations among the solutions,” Physics Letters A, vol. 303, no. 2-3, pp. 197–203, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  38. Y. Yao, “Abundant families of new traveling wave solutions for the coupled Drinfel'd-Sokolov-Wilson equation,” Chaos, Solitons & Fractals, vol. 24, no. 1, pp. 301–307, 2005. View at Publisher · View at Google Scholar · View at Scopus