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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 647313, 6 pages
Exact Solutions and Conservation Laws of a Two-Dimensional Integrable Generalization of the Kaup-Kupershmidt 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 14 November 2012; Accepted 25 November 2012
Academic Editor: Fazal M. Mahomed
Copyright © 2013 Abdullahi Rashid Adem 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.
- M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.
- R. Hirota, “Exact solution of the korteweg-de vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.
- M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters A, vol. 199, no. 3-4, pp. 169–172, 1995.
- Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,” Physics Letters A, vol. 309, no. 5-6, pp. 387–396, 2003.
- G. B. Zheng, B. Liu, Z. J. Wang, and S. K. Zheng, “Variational principle for nonlinear magneto-electroe-lastodynamics with finite displacement by He's semi-inverse method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1523–1526, 2009.
- A. M. 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.
- A. M. Wazwaz, “New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1642–1650, 2006.
- P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993.
- M. Jimbo and T. Miwa, “Solitons and infinite dimensional lie algebras,” Publications of the RIMS, Kyoto University, vol. 19, no. 3, pp. 943–1001, 1983.
- B. G. Konopelchenko and V. G. Dubrovsky, “Some new integrable nonlinear evolution equations in dimensions,” Physics Letters A, vol. 102, no. 1-2, pp. 15–17, 1984.
- V. G. Dubrovsky and Y. V. Lisitsyn, “The construction of exact solutions of two-dimensional integrable generalizations of Kaup-Kuperschmidt and Sawada-Kotera equations via ∂-dressing method,” Physics Letters A, vol. 295, no. 4, pp. 198–207, 2002.
- V. G. Dubrovsky, A. V. Topovsky, and M. Y. Basalaev, “New exact solutions of two-dimensional integrable equations using the ∂-dressing method,” Theoretical and Mathematical Physics, vol. 167, no. 3, pp. 725–739, 2011.
- N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla,USA, 1994–1996.
- G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
- 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, vol. 328, no. 1573, 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.
- S. C. Anco and G. W. Bluman, “Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications,” European Journal of Applied Mathematics, vol. 13, no. 5, pp. 545–566, 2002.
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, NY, USA, 7th edition, 2007.