About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2014 (2014), Article ID 672679, 8 pages
http://dx.doi.org/10.1155/2014/672679
Research Article

Symmetries, Traveling Wave Solutions, and Conservation Laws of a -Dimensional Boussinesq Equation

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Received 5 April 2014; Accepted 15 June 2014; Published 2 July 2014

Academic Editor: Xiao-Yan Gu

Copyright © 2014 Letlhogonolo Daddy Moleleki and Chaudry Masood Khalique. 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. Boussinesq, “Thorie des Ondes et Des Remous Qui se Propagent le Long d’un Canal Rectangulaire Horizontal, en Communiquant au Liquidecontenudansce Canal des Uitessessensiblementpareilles de la Surface au Fond,” Journal de Mathématiques Pures et Appliquées, vol. 17, pp. 55–158, 1872.
  2. G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, NY, USA, 1974. View at MathSciNet
  3. Y. Chen, Z. Yan, and H. Zhang, “New explicit solitary wave solutions for {$(2+1)$}-dimensional Boussinesq equation,” Physics Letters A, vol. 307, no. 2-3, pp. 107–113, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. C. Liu and Z. Dai, “Exact periodic solitary wave solutions for the (2+1)-dimensional Boussinesq equation,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 444–450, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. A. M. Wazwaz, “Non-integrable variants of Boussinesq equation with two solitons,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 820–825, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. G. Imed and B. Abderrahmen, “Numerical solution of the (2+1)-dimensional Boussinesq equation with Initial Condition by Homotopy Perturbation Method,” Applied Mathematical Sciences, vol. 6, pp. 5993–6002, 2012.
  7. L. D. Moleleki and C. M. Khalique, “Solutions and conservation laws of a (2+1)-dimensional Boussinesq equation,” Abstract and Applied Analysis, vol. 2013, Article ID 548975, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. W. Yong-Qi, “Periodic wave solution to the (3 + 1)-dimensional Boussinesq equation,” Chinese Physics Letters, vol. 25, no. 8, pp. 2739–2742, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. M. S. Bruzon and M. L. Gandarias, “Symmetries for a family of Boussinesq equations with nonlinear dispersion,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3250–3257, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. S. Bruzon, “Exact solutions for a generalized Boussinesq equation,” Theoretical and Mathematical Physics, vol. 159, no. 3, pp. 778–785, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. L. Gandarias and M. S. Bruzon, “Classical and nonclassical symmetries of a generalized Boussinesq equation,” Journal of Nonlinear Mathematical Physics, vol. 5, no. 1, pp. 8–12, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. B. Muatjetjeja and C. M. Khalique, “Conservation laws for a variable coefficient variant Boussinesq system,” Abstract and Applied Analysis, vol. 2014, Article ID 169694, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  13. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  14. C. Gu, H. Hu, and Z. Zhou, Darboux Transformation in Soliton Theory and Its Geometric Applications, Springer, Dordrecht, The Netherlands, 2005.
  15. A. 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 · View at MathSciNet · View at Scopus
  16. R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Liu, Z. Fu, 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
  18. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993. View at MathSciNet
  19. N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1, CRC Press, Boca Raton, Fla, USA, 1994.
  20. C. M. Khalique and A. R. Adem, “Exact solutions of a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation using Lie symmetry analysis,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2849–2854, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1217–1231, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. C. M. Khalique, “On the solutions and conservation laws of the (1+1)-dimensional higher-order Broer-Kaup system,” Boundary Value Problems, vol. 2013, article 41, 18 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. N. H. Ibragimov, “A new conservation theorem,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 311–328, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. A. Sjöberg, “Double reduction of {PDE}s from the association of symmetries with conservation laws with applications,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 608–616, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. H. Bokhari, A. Y. Al-Dweik, A. H. Kara, F. M. Mahomed, and F. D. Zaman, “Double reduction of a nonlinear (2+1) wave equation via conservation laws,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1244–1253, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. G. L. Caraffini and M. Galvani, “Symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1474–1484, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, pp. 467–490, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society A, vol. 328, pp. 153–183, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  30. R. J. Knops and C. A. Stuart, “Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity,” Archive for Rational Mechanics and Analysis, vol. 86, no. 3, pp. 233–249, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhuser, Basel, Switzerland, 1992. View at MathSciNet
  32. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1996. View at Publisher · View at Google Scholar · View at MathSciNet