The Traveling Wave Solutions for Some Nonlinear PDEs in Mathematical Physics

Khaled A. Gepreel, Saleh Omran, Sayed K. Elagan

.

DOI: 10.4236/am.2011.23040                

Abstract

In the present article, we construct the exact traveling wave solutions of some nonlinear PDEs in the mathematical physics via (1 + 1) dimensional Kaup Kupershmit equation, the (1 + 1) dimensional seventh order KdV equation and (1 + 1) dimensional Kersten-Krasil Shchik equations by using the modified truncated expansion method. New exact solutions of these equations are found.

Keywords

Conjugate Modified Truncated Expansion Method, Traveling Wave Solutions, Nonlinear Evolution Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Weiss, M. Tabor and G. Carnevalle, “The Painleve Property for Partial Differential Equations,” Journal of Mathematical Physics, Vol. 24, No. 3, 1983, pp. 522-526. doi:10.1063/1.525721
[2]	N. A. Kudryashov, “Exact Soliton Solutions of the Generalized Evolution Equation of Wave Dynamics,” Journal of Applied Mathematics and Mechanics, Vol. 52, No. 3, 1988, pp. 361-365. doi:10.1016/0021-8928(88)90090-1
[3]	J. Weiss, “The Panleve Property for Partial Differential Equations. II: Backlund Transformation, Lax Pairs, and the Schwarzian Derivative,” Journal of Mathematical Physics, Vol. 24, No. 6, 1983, pp. 1405-1413. doi:10.1063/1.525875
[4]	N. A. Kudryashov, “Exact Solutions of the Generalized Kuramoto—Sivashinsky Equation,” Physics Letters A, Vol. 147, No. 5-6, 1990, pp. 287-291. doi:10.1016/0375-9601(90)90449-X
[5]	N. A. Kudryashov and N. B. Loguinova, “Extended Simpliest Equation Method for Nonlinear Differential Equations,” Applied Mathematics and Computation, Vol. 205, No. 1, 2008, pp. 396-402. doi:10.1016/j.amc.2008.08.019
[6]	E. J. Parkes and B. R. Duffy, “An Automated Tanh-function Method for Finding Solitary Wave Solutions to Nonlinear Evolution Equations,” Computer Physics Communications, Vol. 98, No. 3, 1996, pp. 288-300. doi:10.1016/0010-4655(96)00104-X
[7]	N. A. Kudryashov and M. V. Demina, “Polygons of Differential Equations for Finding Exact Solutions,” Chaos, Solitons & Fractals, Vol. 33, No. 5, 2007. pp. 1480-1496. doi:10.1016/j.chaos.2006.02.012
[8]	P. A. Clarkson and M. D. Kruskal, “New Similarity Reductions of the Boussinesq Equation,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, pp. 2201-2213. doi:10.1063/1.528613
[9]	P. N. Ryabov, “Exact Solutions of the Kudryashov-Sinelshchikov Equation,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3585-3590. doi:10.1016/j.amc.2010.09.003
[10]	N. A. Kudryashov, “Analytical Theory of Nonlinear Differential Equations,” Institute of Computer Investigations, Moscow, 2004.
[11]	M. Musette and C. Verhoeven, “Nonlinear Superposition Formula for the Kaup-Kupershmidt Partial Differential Equation,” Physica D, Vol. 144, No. 1-2, 2000, pp. 211-220. doi:10.1016/S0167-2789(00)00081-6
[12]	A. H. Salas, “Computing Exact Solutions to a Generalized Lax Seventh-Order Forced KdV Equation,” Applied Mathematics and Computation, Vol. 216, No. 8, 2010, pp. 2333-2338. doi:10.1016/j.amc.2010.03.078
[13]	E. M. E. Zayed and K. A. Gepreel, “New Applications of an Improved(G′/G)-Expansion Method to Constract the Exact Solutions of Nonlinear PDEs,” International Journal of nonlinear Science and Numerical Simulation, Vol. 11, No. 4, 2010, pp. 273-283.
[14]	A. K. Kalkanli, S. Y. Sakovich and T. Yurdusen, “Integrability of Kersten-Krasil’shchik Coupled KdV-mKdV Equations: Singularity Analysis and Lax Pair,” Journal of Mathematical Physics, Vol. 44, No. 4, 2003, pp. 1703-1708. doi:10.1063/1.1558903