## Abstract and Applied Analysis

### Classification of Exact Solutions for Some Nonlinear Partial Differential Equations with Generalized Evolution

#### Abstract

We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 478531, 16 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174069

Digital Object Identifier
doi:10.1155/2012/478531

Mathematical Reviews number (MathSciNet)
MR2965451

Zentralblatt MATH identifier
1247.35124

#### Citation

Pandir, Yusuf; Gurefe, Yusuf; Kadak, Ugur; Misirli, Emine. Classification of Exact Solutions for Some Nonlinear Partial Differential Equations with Generalized Evolution. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 478531, 16 pages. doi:10.1155/2012/478531. https://projecteuclid.org/euclid.aaa/1365174069

#### References

• 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.
• W. Malfliet and W. Hereman, “The tanh method. I. Exact solutions of nonlinear evolution and wave equations,” Physica Scripta, vol. 54, no. 6, pp. 563–568, 1996.
• H. X. Wu and J. H. He, “Exp-function method and its application to nonlinear equations,” Chaos, Solitons and Fractals, vol. 30, pp. 700–708, 2006.
• E. Misirli and Y. Gurefe, “Exp-function method for solving nonlinear evolution equations,” Mathematical & Computational Applications, vol. 16, no. 1, pp. 258–266, 2011.
• Y. Gurefe and E. Misirli, “Exp-function method for solving nonlinear evolution equations with higher order nonlinearity,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2025–2030, 2011.
• M. Wang, X. Li, and J. Zhang, “The $({G}^{\prime }/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters. A, vol. 372, no. 4, pp. 417–423, 2008.
• Y. Gurefe and E. Misirli, “New variable separation solutions of two-dimensional Burgers system,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9189–9197, 2011.
• Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics,” Pramana, vol. 77, pp. 1023–1029, 2011.
• Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of an irrational trial equation method to high-dimensional nonlinear evolution equations,” Journal of Advanced Mathematical Studies, vol. 5, pp. 41–47, 2012.
• C. S. Liu, “Trial equation method and its applications to nonlinear evolution equations,” Acta Physica Sinica, vol. 54, no. 6, pp. 2505–2509, 2005.
• C. S. Liu, “Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications,” Communications in Theoretical Physics, vol. 45, pp. 219–223, 2006.
• C. S. Liu, “A new trial equation method and its applications,” Communications in Theoretical Physics, vol. 45, pp. 395–397, 2006.
• C. Y. Jun, “Classification of traveling wave solutions to the Vakhnenko equations,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3987–3996, 2011.
• C. Y. Jun, “Classification of traveling wave solutions to the modified form of the Degasperis-Procesi equation,” Mathematical and Computer Modelling, vol. 56, pp. 43–48, 2012.
• C.-S. Liu, “Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations,” Computer Physics Communications, vol. 181, no. 2, pp. 317–324, 2010.
• M. M. Kabir, “Analytic solutions for a nonlinear variant of the (2+1) dimensional Camassa Holm KP equation,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 1566–1577, 2011.
• W. Zhang, Q. Chang, and B. Jiang, “Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order,” Chaos, Solitons and Fractals, vol. 13, no. 2, pp. 311–319, 2002.