About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 271960, 6 pages
http://dx.doi.org/10.1155/2014/271960
Research Article

Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System

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 22 January 2014; Accepted 4 March 2014; Published 27 March 2014

Academic Editor: Baojian Hong

Copyright © 2014 Catherine Matjila 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. Z. Wen, Z. Liu, and M. Song, “New exact solutions for the classical Drinfel'd-Sokolov-Wilson equation,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2349–2358, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R.-x. Yao and Z.-b. Li, “New exact solutions for three nonlinear evolution equations,” Physics Letters A, vol. 297, no. 3-4, pp. 196–204, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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
  4. 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
  5. E. Fan, “An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations,” Journal of Physics A: Mathematical and General, vol. 36, no. 25, pp. 7009–7026, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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
  7. 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
  8. X.-Q. Zhao and H.-Y. Zhi, “An improved F-expansion method and its application to coupled Drinfel'd-Sokolov-Wilson equation,” Communications in Theoretical Physics, vol. 50, no. 2, pp. 309–314, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  9. 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 Zentralblatt MATH · View at MathSciNet
  10. Z. Y. Yan and H. Q. Zhang, “On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics,” Applied Mathematics and Mechanics, vol. 21, no. 4, pp. 383–388, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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
  12. A. M. Wazwaz, “Compactons and solitary patterns structures for variants of the KdV and the KP equations,” Applied Mathematics and Computation, vol. 139, no. 1, pp. 37–54, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. 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 Zentralblatt MATH · View at MathSciNet
  14. A. M. Wazwaz, “The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the K(n,n)-Burger equations,” Physica D, vol. 213, no. 2, pp. 147–151, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  15. 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
  16. 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
  17. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 328, pp. 153–183, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  18. 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
  19. R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, Switzerland, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  20. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Germany, 1996. View at MathSciNet
  21. A. Sjöberg, “Double reduction of PDEs 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 Zentralblatt MATH · View at MathSciNet
  22. 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
  23. 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
  24. E. Noether, “Invariante variationsprobleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu G öttingen, Mathematisch-Physikalische Klasse, Heft, vol. 2, pp. 235–257, 1918.
  25. P. S. Laplace, Traite de Mecanique Celeste, vol. 1, Paris, France, 1978, (English translation Celestial Mechanics, New York, NY, USA, 1966).
  26. H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung, vol. 17a, pp. 129–132, 1962. View at MathSciNet
  27. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  28. S. C. Anco and G. Bluman, “Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,” European Journal of Applied Mathematics, vol. 13, no. 5, pp. 545–566, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. A. H. Kara and F. M. Mahomed, “Relationship between symmetries and conservation laws,” International Journal of Theoretical Physics, vol. 39, no. 1, pp. 23–40, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. H. Kara and F. M. Mahomed, “Noether-type symmetries and conservation laws via partial Lagrangians,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367–383, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. Naz, F. M. Mahomed, and T. Hayat, “Conservation laws for third-order variant Boussinesq system,” Applied Mathematics Letters, vol. 23, no. 8, pp. 883–886, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet