On Exact Traveling Wave Solutions for (1 + 1) Dimensional Kaup-Kupershmidt Equation
Dahe Feng, Kezan Li
.
DOI: 10.4236/am.2011.26100   PDF    HTML     5,993 Downloads   10,990 Views   Citations

Abstract

In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.

Share and Cite:

D. Feng and K. Li, "On Exact Traveling Wave Solutions for (1 + 1) Dimensional Kaup-Kupershmidt Equation," Applied Mathematics, Vol. 2 No. 6, 2011, pp. 752-756. doi: 10.4236/am.2011.26100.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, London, 1991. doi:10.1017/CBO9780511623998
[2] H. Z. Liu, J. B. Li and L. Liu, “Lie Symmetry Analysis, Optimal Systems and Exact Solutions to the Fifth-Order KdV Types of Equations,” Journal of Mathematical Analysis and Applications, Vol. 368, No. 2, 2011, pp. 551-558. doi:10.1016/j.jmaa.2010.03.026
[3] H. Z. Liu and J. B. Li, “Lie Symmetry Analysis and Exact Solutions for the Short Pulse Equation,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 71, No. 5-6, 2009, pp. 2126-2133. doi:10.1016/j.na.2009.01.075
[4] A. Y. Chen and J. B. Li, “Single Peak Solitary Wave Solutions for the Osmosis K(2,2) Equation under Inhomogeneous Boundary Condition,” Journal of Mathematical Analysis and Applications, Vol. 369, No. 2, 2010, pp. 758-766. doi:10.1016/j.jmaa.2010.04.018
[5] D. H. Feng and J. B. Li, “Exact Explicit Traveling Wave Solutions for the (n + 1)-Dimensional Φ6 Field Model,” Physics Letters A, Vol. 369, No. 4, 2007, pp. 255-261. doi:10.1016/j.physleta.2007.04.088
[6] J. W. Shen, W. Xu and Y. Xu, “Traveling Wave Solutions in the Generalized Hirota-Satsuma Coupled KdV System,” Applied Mathematics and Computation, Vol. 161, No. 2, 2005, pp. 365-383. doi:10.1016/j.amc.2003.12.033
[7] A. M. Wazwaz, “A Sine-Cosine Method for Handling Nonlinear Wave Equations,” Mathematical and Computer Modelling, Vol. 40, No. 5-6, 2004, pp. 499-508. doi:10.1016/j.mcm.2003.12.010
[8] A. M. Wazwaz, “Solitons and Periodic Solutions for the Fifth-Order KdV Equation,” Applied Mathematics Letters, Vol. 19, No. 11, 2006, pp. 1162-1167. doi:10.1016/j.aml.2005.07.014
[9] A. M. Wazwaz, “Analytic Study on Nonlinear Variant of the RLW and the PHI-Four Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 12, No. 3, 2007, pp. 314-327. doi:10.1016/j.cnsns.2005.03.001
[10] Z. Y. Yan, “New Explicit Traveling Wave Solutions for Two New Integrable Coupled Nonlinear Evolution Equations,” Physics Letters A, Vol. 292, No. 1-2, 2001, pp. 100-106. doi:10.1016/S0375-9601(01)00772-1
[11] M. A. Abdou, “New Solitons and Periodic Wave Solutions for Nonlinear Physical Models,” Nonlinear Dynamics, Vol. 52, No. 1-2, 2008, pp. 129-136. doi:10.1007/s11071-007-9265-7
[12] M. L. Wang, “Exact Solutions for a Compound KdV- Burgers Equation,” Physics Letters A, Vol. 213, No. 5-6, 1996, pp. 279-287. doi:10.1016/0375-9601(96)00103-X
[13] X. J. Deng, M. Zhao and X. Li, “Traveling Wave Solutions for a Nonlinear Variant of the PHI-Four Equation,” Mathematical and Computer Modelling, Vol. 49, No. 3-4, 2009, pp. 617-622. doi:10.1016/j.mcm.2008.03.011
[14] E. G. Fan, “Uniformly Constructing a Series of Explicit Exact Solutions to Nonlinear Equations in Mathematical Physics,” Chaos, Solitons and Fractals, Vol. 16, No. 5, 2005, pp. 819-839. doi:10.1016/S0960-0779(02)00472-1
[15] S. Zhang and H. Q. Zhang, “Fan Sub-Equation Method for Wick-Type Stochastic Partial Differential Equations,” Physics Letters A, Vol. 374, No. 41, 2010, pp. 4180-4187. doi:10.1016/j.physleta.2010.08.023
[16] D. H. Feng and G. X. Luo, “The Improved Fan Sub-Equation Method and Its Application to the SK Equation,” Applied Mathematics and Computation, Vol. 215, No. 5, 2009, pp. 1949-1967. doi:10.1016/j.amc.2009.07.045
[17] E. Yomba, “The Extended Fan’s Sub-Equation Method and Its Application to KdV-MKdV, BKK and Variant Boussinesq Equations,” Physics Letters A, Vol. 336, No. 6, 2005, pp. 463-476. doi:10.1016/j.physleta.2005.01.027
[18] K. A. Gepreel, S. Omran and S. K. Elagan, “The Traveling Wave Solutions for Some Nonlinear PDEs in Mathematical Physics,” Applied Mathematics, Vol. 2, No. 3, 2011, pp. 343-347. doi:10.4236/am.2011.23040
[19] B. A. Kupershmidt, “A Super Korteweg-de Vries Equation: An Integrable System,” Physics Letters A, Vol. 102, No. 5-6, 1984, pp. 213-215. doi:10.1016/0375-9601(84)90693-5
[20] 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
[21] U. Goktas and W. Hereman, “Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations,” Journal of Symbolic Computation, Vo. 24, No. 5, 1997, pp. 591-622. doi:10.1006/jsco.1997.0154
[22] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R. S. Martino and J. C. Miller, “Symbolic Computation of Exact Solutions Expressible in Hyperbolic and Elliptic Functions for Nonlinear PDEs,” Journal of Symbolic Computation, Vol. 37, No. 6, 2004, pp. 699-705. doi:10.1016/j.jsc.2003.09.004

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.