- 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 2012 (2012), Article ID 489043, 11 pages
The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation
1School of Computer Science and Educational Software, Guangzhou University, Guangzhou 510006, China
2School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Received 25 October 2011; Accepted 14 November 2011
Academic Editor: Shaher M. Momani
Copyright © 2012 Yong Huang and Yadong Shang. 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, vol. 149, Cambridge University Press, Cambridge, UK, 1991.
- B. Grammaticos, A. Ramani, and J. Hietarinta, “A search for integrable bilinear equations: the Painlevé approach,” Journal of Mathematical Physics, vol. 31, no. 11, pp. 2572–2578, 1990.
- G. B. Whitham, “Comments on periodic waves and solitons,” IMA Journal of Applied Mathematics, vol. 32, no. 1–3, pp. 353–366, 1984.
- G. B. Whitham, “On shocks and solitary waves,” Scripps Institution of Oceanography Reference Series, pp. 91–124, 1991.
- J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983.
- J. D. Gibbon, A. C. Newell, M. Tabor, and Y. B. Zeng, “Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations,” Nonlinearity, vol. 1, no. 3, pp. 481–490, 1988.
- J. Weiss, “Bäcklund transformation and the Painlevé property,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
- J. Hietarinta, “Hirota's bilinear method and partial integrability,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
- N. G. Berloff and L. N. Howard, “Solitary and periodic solutions of nonlinear nonintegrable equations,” Studies in Applied Mathematics, vol. 99, no. 1, pp. 1–24, 1997.
- M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996.
- M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996.
- M. L. Wang, Y. B. Zhou, and H. Q. Zhang, “A nonlinear transformation of the shallow water wave equations and its application,” Advances in Mathematics, vol. 28, no. 1, pp. 72–75, 1999.
- E. G. Fan and H. Q. Zhang, “A new approach to Bäcklund transformations of nonlinear evolution equations,” Applied Mathematics and Mechanics, vol. 19, no. 7, pp. 645–650, 1998.
- Y. D. Shang, “Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1286–1297, 2007.
- Y. D. Shang, J. Qin, Y. Huang, and W. Yuan, “Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma-Tasso-Olver equation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 532–538, 2008.
- C. Roman and P. Oleksii, “New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations,” Journal of Physics A, vol. 40, no. 33, pp. 10049–10070, 2007.
- J. F. Zhang, “Exact and explicit solitary wave solutions to some nonlinear equations,” International Journal of Theoretical Physics, vol. 35, no. 8, pp. 1793–1798, 1996.
- A. M. Wazwaz, “Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 639–656, 2005.
- R. Cherniha, “New Q symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 783–799, 2007.
- J. H. Merkin, R. A. Satnoianu, and S. K. Scott, “Travelling waves in a differential flow reactor with simple autocatalytic kinetics,” Journal of Engineering Mathematics, vol. 33, no. 2, pp. 157–174, 1998.