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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 340564, 11 pages
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.
- 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).
- 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.
- 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.
- 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.
- H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, UK, 1989.
- P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986.
- 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.
- A. Sjöberg, “On double reductions from symmetries and conservation laws,” Nonlinear Analysis: Real World Applications, vol. 10, no. 6, pp. 3472–3477, 2009.
- 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.
- 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.
- 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.
- H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung A, vol. 17, pp. 129–132, 1962.
- 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.
- I. Aslan, “Generalized solitary and periodic wave solutions to a ()-dimensional Zakharov-Kuznetsov equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1421–1429, 2010.
- 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.
- A.-M. Wazwaz, “Solitons and singular solitons for the Gardner-KP equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 162–169, 2008.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.