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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 271960, 6 pages
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- X.-Q. Zhao and H.-Y. Zhi, “An improved -expansion method and its application to coupled Drinfel'd-Sokolov-Wilson equation,” Communications in Theoretical Physics, vol. 50, no. 2, pp. 309–314, 2008.
- 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.
- 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.
- M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters A, vol. 199, no. 3-4, pp. 169–172, 1995.
- 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.
- E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
- A. M. Wazwaz, “The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the -Burger equations,” Physica D, vol. 213, no. 2, pp. 147–151, 2006.
- M. Wang, X. Li, and J. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
- P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, pp. 467–490, 1968.
- 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.
- 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.
- R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, Switzerland, 1992.
- E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Germany, 1996.
- 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. H. Bokhari, A. Y. Al-Dweik, A. H. Kara, F. M. Mahomed, and F. D. Zaman, “Double reduction of a nonlinear wave equation via conservation laws,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1244–1253, 2011.
- 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.
- E. Noether, “Invariante variationsprobleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu G öttingen, Mathematisch-Physikalische Klasse, Heft, vol. 2, pp. 235–257, 1918.
- P. S. Laplace, Traite de Mecanique Celeste, vol. 1, Paris, France, 1978, (English translation Celestial Mechanics, New York, NY, USA, 1966).
- H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung, vol. 17a, pp. 129–132, 1962.
- P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993.
- 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.
- 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, “Noether-type symmetries and conservation laws via partial Lagrangians,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367–383, 2006.
- 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.