Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 904252, 5 pages
http://dx.doi.org/10.1155/2014/904252
Research Article

Conservation Laws for a Degasperis Procesi Equation and a Coupled Variable-Coefficient Modified Korteweg-de Vries System in a Two-Layer Fluid Model via the Multiplier Approach

1Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
2Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt

Received 21 May 2014; Revised 20 September 2014; Accepted 16 October 2014; Published 13 November 2014

Academic Editor: Athanasios N. Yannacopoulos

Copyright © 2014 E. Osman 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. P. S. Laplace, Traité de Mécanique Céleste, vol. 1, Paris, 1798. English Translation, Celestrial Mechanics, New York, NY, USA, 1966.
  2. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. C. Anco and G. 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 MathSciNet · View at Scopus
  4. 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 MathSciNet · View at Scopus
  5. T. Wolf, “A comparison of four approaches to the calculation of conservation laws,” European Journal of Applied Mathematics, vol. 13, no. 2, pp. 129–152, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. T. Wolf, A. Brand, and M. Mohammadzadeh, “Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions,” Journal of Symbolic Computation, vol. 27, no. 2, pp. 221–238, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Ü. Göktaş and W. Hereman, “Symbolic computation of conserved densities for systems of nonlinear evolution equations,” Journal of Symbolic Computation, vol. 24, no. 5, pp. 591–621, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman, and B. M. Herbst, “Direct methods and symbolic software for conservation laws of nonlinear equations,” in Advances of Nonlinear Waves and Symbolic Computation, Z. Yan, Ed., chapter 2, pp. 19–79, Nova Science Publishers, New York, NY, USA, 2009. View at Google Scholar
  9. W. Hereman, M. Colagrosso, R. Sayers et al., “Continuous and discrete homotopy operators and the computation of conservation laws,” in Differential Equations with Symbolic Computation, D. Wang and Z. Zheng, Eds., pp. 249–285, Birkhäuser, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  10. W. Hereman, “Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions,” International Journal of Quantum Chemistry, vol. 106, no. 1, pp. 278–299, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. A. F. Cheviakov, “GeM software package for computation of symmetries and conservation laws of differential equations,” Computer Physics Communications, vol. 176, no. 1, pp. 48–61, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Y.-F. Wang, B. Tian, M. Li, P. Wang, and M. Wang, “Integrability and soliton-like solutions for the coupled higher-order nonlinear Schrödinger equations with variable coefficients in inhomogeneous optical fibers,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 6, pp. 1783–1791, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z. Hui-Ling, T. Bo, W. Yu-Feng, Z. H. Hui, and S. Wen-Rong, “Dynamic behavior of the quantum Zakharov-Kuznetsov equations in dense quantum magnetoplasmas,” Physics of Plasmas, vol. 21, Article ID 012304, 2014. View at Google Scholar
  14. Y.-J. Shen, Y.-T. Gao, D.-W. Zuo, Y.-H. Sun, Y.-J. Feng, and L. Xue, “Nonautonomous matter waves in a spin-1 Bose-Einstein condensate,” Physical Review E, vol. 89, no. 6, Article ID 062915, 12 pages, 2014. View at Publisher · View at Google Scholar
  15. H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssatzen,” Zeitschrift für Naturforschung, vol. 17A, pp. 129–132, 1962. View at Google Scholar · View at MathSciNet
  16. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta, Eds., pp. 23–27, World Scientific Publishers, Singapore, 1999. View at Google Scholar · View at MathSciNet
  18. S.-H. Zhu, Y.-T. Gao, X. Yu, Z.-Y. Sun, X.-L. Gai, and D.-X. Meng, “Painlevé property, soliton-like solutions and complexitons for a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 295–307, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Y. Gao and X.-Y. Tang, “A coupled variable coefficient modified KdV equation arising from a two-layer fluid system,” Communications in Theoretical Physics, vol. 48, no. 6, pp. 961–970, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus