About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 647313, 6 pages
http://dx.doi.org/10.1155/2013/647313
Research Article

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.

Linked References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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. View at Publisher · View at Google Scholar · View at Scopus
  3. 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
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Scopus
  6. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  9. 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. View at Publisher · View at Google Scholar
  10. B. G. Konopelchenko and V. G. Dubrovsky, “Some new integrable nonlinear evolution equations in (2+1) dimensions,” Physics Letters A, vol. 102, no. 1-2, pp. 15–17, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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. View at Publisher · View at Google Scholar · View at Scopus
  13. N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla,USA, 1994–1996.
  14. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  15. 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
  16. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society, vol. 328, no. 1573, pp. 153–183, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  17. 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
  18. R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, Switzerland, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  19. E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Germany, 1996. View at MathSciNet
  20. 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
  21. 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
  22. 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
  23. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, NY, USA, 7th edition, 2007. View at MathSciNet