Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2015 (2015), Article ID 369294, 7 pages
http://dx.doi.org/10.1155/2015/369294
Research Article

Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations

1Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia
2Mathematics Department, Faculty of Science, Beni-Suef University, Egypt
3Mathematics Department, College of Arts and Science, Taif University, Ranyah, Saudi Arabia

Received 24 August 2014; Accepted 1 November 2014

Academic Editor: Yasir Khan

Copyright © 2015 A. R. Seadawy and K. El-Rashidy. 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. J. M. Burgers, “A mathematical model illustrating the theory of turbulence,” in Advances in Applied Mechanics, pp. 171–199, Academic Press, New York, NY, USA, 1948. View at Google Scholar
  2. C. A. J. Fletcher, “Generating exact solutions of the two-dimensional Burgers' equations,” International Journal for Numerical Methods in Fluids, vol. 3, no. 3, pp. 213–216, 1983. View at Publisher · View at Google Scholar · View at Scopus
  3. J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,” Quarterly of Applied Mathematics, vol. 9, pp. 225–236, 1951. View at Google Scholar · View at MathSciNet
  4. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  5. G. Bluman and S. Kumei, Symmetries and Differenti al Equations, Springer, New York, NY, USA, 1986. View at MathSciNet
  6. J. Zhang, F. Wu, and J. Shi, “Simple soliton solution method for the combined KdV and MKdV equation,” International Journal of Theoretical Physics, vol. 39, no. 6, pp. 1697–1702, 2000. View at Publisher · View at Google Scholar
  7. A. R. Seadawy, “New exact solutions for the KdV equation with higher order nonlinearity by using the variational method,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3741–3755, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. A. R. Seadawy, “Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma,” Computers & Mathematics with Applications, vol. 67, no. 1, pp. 172–180, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. P. Agrawal, Applications of Nonlinear Fiber Optics, Elsevier, 2008.
  10. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. H. Khater, M. A. Helal, and A. R. Seadawy, “General soliton solutions of ndimensional nonlinear Schrödinger equation,” IL Nuovo Cimento B, vol. 115, pp. 1303–1312, 2000. View at Google Scholar
  12. A. R. Seadawy, “Exact solutions of a two-dimensional nonlinear Schrödinger equation,” Applied Mathematics Letters, vol. 25, no. 4, pp. 687–691, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. Z. Yan and H. 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 MathSciNet · View at Scopus
  15. C. Rogers and W. F. Shadwick, Backlund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982. View at MathSciNet
  16. R. Hirota, “Exact solution of the korteweg-de vries equation for multiple Collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. J. H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 235–236, 1997. View at Publisher · View at Google Scholar · View at Scopus
  18. A. S. Arife and A. Yildirim, “New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step,” World Applied Sciences Journal, vol. 13, no. 10, pp. 2186–2190, 2011. View at Google Scholar
  19. M. A. Helal and A. R. Seadawy, “Variational method for the derivative nonlinear Schrödinger equation with computational applications,” Physica Scripta, vol. 80, no. 3, Article ID 035004, pp. 350–360, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. S. T. Mohyud-Din, “Modified variational iteration method for integro-differential equations and coupled systems,” Zeitschrift für Naturforschung A, vol. 65, no. 4, pp. 277–284, 2010. View at Google Scholar
  21. W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” The American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Bekir and A. C. Cevikel, “Solitary wave solutions of two nonlinear physical models by tanh-coth method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1804–1809, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. A. R. Seadawy and K. El-Rashidy, “Traveling wave solutions for some coupled nonlinear evolution equations,” Mathematical and Computer Modelling, vol. 57, no. 5-6, pp. 1371–1379, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. A. R. Seadawy, “Traveling wave solutions of the Boussinesq and generalized fifth order KdV equations by using the direct algebraic method,” Applied Mathematical Sciences, vol. 6, pp. 4081–4090, 2012. View at Google Scholar
  25. A. H. Salas and C. A. Gomez, “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. W. X. Ma and B. Fuchssteiner, “Explicit and exact solutions to a Kolmogorov-Petrovskii-PISkunov equation,” International Journal of Non-Linear Mechanics, vol. 31, no. 3, pp. 329–338, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. H. Naher, F. A. Abdullah, and M. A. Akbar, “The exp-function method for new exact solutions of the nonlinear partial differential equations,” International Journal of Physical Sciences, vol. 6, no. 29, pp. 6706–6716, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. I. Aslan and V. Marinakis, “Some remarks on exp-function method and its applications,” Communications in Theoretical Physics, vol. 56, no. 3, pp. 397–403, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. A. Yıldırım and Z. Pınar, “Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1873–1880, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. A.-M. Wazwaz, “A new (2+1)-dimensional Korteweg-de Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation,” Physica Scripta, vol. 84, no. 3, Article ID 035010, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. 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 MathSciNet · View at Scopus
  32. D. Kaya, “An explicit solution of coupled viscous Burgers' equation by the decomposition method,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 11, pp. 675–680, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet