About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 340564, 11 pages
http://dx.doi.org/10.1155/2013/340564
Research Article

Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
2Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore 54000, Pakistan
3Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, Opposite Sector U, DHA, Lahore Cantt 54792, Pakistan

Received 26 May 2013; Accepted 16 July 2013

Academic Editor: Teoman Özer

Copyright © 2013 R. Naz 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. E. Noether, “Invariante Variationsprobleme,” Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, vol. 2, pp. 235–257, 1918, (English translation in Transport Theory and Statistical Physiscs, vol. 1, no. 3, 186–207, 1971). View at MathSciNet
  2. A. H. Kara and F. M. Mahomed, “Action of Lie Backlund symmetries on conservation laws,” in Proceedings of the 10th International Conference on Modern Group Analysis, vol. 7, Nordfjordeid, Norway, 1997.
  3. A. Sjöberg and F. M. Mahomed, “Non-local symmetries and conservation laws for one-dimensional gas dynamics equations,” Applied Mathematics and Computation, vol. 150, no. 2, pp. 379–397, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Sjöberg and F. M. Mahomed, “The association of non-local symmetries with conservation laws: applications to the heat and Burgers' equations,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1098–1108, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  5. H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, UK, 1989. View at MathSciNet
  6. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  7. 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
  8. A. Sjöberg, “On double reductions from symmetries and conservation laws,” Nonlinear Analysis: Real World Applications, vol. 10, no. 6, pp. 3472–3477, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. H. Bokhari, A. Y. Al-Dweik, F. D. Zaman, A. H. Kara, and F. M. Mahomed, “Generalization of the double reduction theory,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3763–3769, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Naz, M. D. Khan, and I. Naeem, “Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 4, pp. 826–834, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Naz, F. M. Mahomed, and D. P. Mason, “Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 212–230, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung A, vol. 17, pp. 129–132, 1962. View at MathSciNet
  13. R. Naz, D. P. Mason, and F. M. Mahomed, “Conservation laws and conserved quantities for laminar two-dimensional and radial jets,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2641–2651, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. I. Aslan, “Generalized solitary and periodic wave solutions to a (2+1)-dimensional Zakharov-Kuznetsov equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1421–1429, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z. Fu, S. Liu, and S. Liu, “Multiple structures of two-dimensional nonlinear Rossby wave,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 383–390, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A.-M. Wazwaz, “Solitons and singular solitons for the Gardner-KP equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 162–169, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Zhao, W. Xu, H. Jia, and H. Zhou, “Solitary wave solutions for the modified Kadomtsev-Petviashvili equation,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 465–475, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Q. M. Al-Mdallal and M. I. Syam, “Sine-cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation,” Chaos, Solitons and Fractals, vol. 33, no. 5, pp. 1610–1617, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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 Zentralblatt MATH · View at MathSciNet
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. Z. Feng, “The first-integral method to study the Burgers-Korteweg-de Vries equation,” Journal of Physics A, vol. 35, no. 2, pp. 343–349, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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
  23. A. H. Kara and F. M. Mahomed, “A basis of conservation laws for partial differential equations,” Journal of Nonlinear Mathematical Physics, vol. 9, no. 2, pp. 60–72, 2002. View at Publisher · View at Google Scholar · View at MathSciNet