- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 169694, 5 pages
Conservation Laws for a Variable Coefficient Variant Boussinesq System
Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
Received 21 November 2013; Accepted 6 January 2014; Published 12 February 2014
Academic Editor: Hossein Jafari
Copyright © 2014 Ben Muatjetjeja 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.
- J. Boussinesq, “Thorie de l'intumescence liquide, applele onde solitaire ou de translation, se propageant dans un canal rectangulaire,” Comptes Rendus de L'Academie des Sciences, vol. 72, pp. 755–759, 1871.
- R. L. Sachs, “On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy,” Physica D, vol. 30, no. 1-2, pp. 1–27, 1988.
- E. Fan and Y. C. Hon, “A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, no. 3, pp. 559–566, 2003.
- M. Wang, “Solitary wave solution variant Boussinesq equations travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, pp. 559–566, 2003.
- Z. Fu, S. Liu, and S. Liu, “New transformations and new approach to find exact solutions to nonlinear equations,” Physics Letters A, vol. 299, no. 5-6, pp. 507–512, 2002.
- J. F. Zhang, “Multi-solitary wave solutions for variant Boussinesq equations and Kupershmidt equations,” Applied Mathematics and Mechanics, vol. 21, no. 2, pp. 171–175, 2000.
- G.-Q. Xu, Z.-B. Li, and Y.-P. Liu, “Exact solutions to a large class of nonlinear evolution equations,” The Chinese Journal of Physics, vol. 41, no. 3, pp. 232–241, 2003.
- 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.
- P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, no. 5, pp. 467–490, 1968.
- T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society A, 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, Birkhuser, Basel, Switzerland, 2nd edition, 1992.
- E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences, 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 Göttingen, vol. 2, pp. 235–257, 1918.
- N. H. Ibragimov, Ed., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla, USA, 1994–1996.
- 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.