General Solution of Generalized (2+1)–Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –Expansion Method

Abdollah Borhanifar, Reza Abazari
DOI: 10.4236/ajcm.2011.14025                

Abstract

In this work, the (G^,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G^,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Keywords

(G, /G)-Expansion Method, Generalized Kadomtsev-Petviashvili (KP) Equation, Hyperbolic Function Solutions, Trigonometric Function Solutions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	G. T. Liu and T. Y. Fan, “New Applications of Devel- oped Jacobi Elliptic Function Expansion Methods,” Phy- sics Letters A, Vol. 345, No. 1-3, 2005, pp. 161-166. doi:10.1016/j.physleta.2005.07.034
[2]	M. J. Ablowitz and H. Segur, “Solitons and Inverse Scattering Transform,” SIAM, Philadelphia, 1981. doi:10.1137/1.9781611970883
[3]	R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridge, 2004.
[4]	M. L. Wang, “Exact Solutions for a Compound KdV-Burg- ers Equation,” Physics Letters A, Vol. 213, No. 5-6, 1996, pp. 279-287. doi:10.1016/0375-9601(96)00103-X
[5]	J. H. He, “The Homotopy Perturbation Method for Non- linear Oscillators with Discontinuities,” Applied Mathematics and Computation, Vol. 151, No. 1, 2004, pp. 287-292. doi:10.1016/S0096-3003(03)00341-2
[6]	Z. Y. Yan, “An Improved Algebra Method and Its Applications in Nonlinear Wave Equations,” Chaos Solitons & Fractals, Vol. 21, No. 4, 2004, pp. 1013-1021. doi:10.1016/j.chaos.2003.12.042
[7]	G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” Springer-Verlag, New York, 1989.
[8]	G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer, Boston, 1994.
[9]	A. Borhanifar, H. Jafari and S. A. Karimi, “New Solitons and Periodic Solutions for the Kadomtsev-Petviashvili Equa- tion,” The Journal of Nonlinear Science and Applications, Vol. 1, No. 4, 2008, pp. 224-229.
[10]	H. Jafari, A. Borhanifar and S. A. Karimi, “New Solitary Wave Solutions for the Bad Boussinesq and Good Bous- sinesq Equations,” Numerical Methods for Partial Dif- ferential Equations, Vol. 25, No. 5, 2000, pp. 1231-1237.
[11]	A. Borhanifar, M. M. Kabir and L. M. Vahdat, “New Pe- riodic and Soliton Wave Solutions for the Generalized Zak- harov System and (2 + 1)-Dimensional Nizhnik-Novikov- Veselov System,” Chaos Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1646-1654. doi:10.1016/j.chaos.2009.03.064
[12]	A. Borhanifar and M. M. Kabir, “New Periodic and Soli- ton Solutions by Application of Exp-Function Method for Nonlinear Evolution Equations,” Journal of Computa- tional and Applied Mathematics, Vol. 229, No. 1, 2009, pp. 158-167. doi:10.1016/j.cam.2008.10.052
[13]	S. A. El Wakil, M. A. Abdou and A. Hendi, “New Peri- odic Wave Solutions via Exp-Function Method,” Physics Letters A, Vol. 372, No. 6, 2008, pp. 830-840. doi:10.1016/j.physleta.2007.08.033
[14]	A. Boz and A. Bekir, “Application of Exp-Function Me- thod for (3 + 1)-Dimensional Nonlinear Evolution Equa- tions,” Computers & Mathematics with Applications, Vol. 56, No. 5, 2000, pp. 1451-1456.
[15]	H. Zhao and C. Bai, “New Doubly Periodic and Multiple Soliton Solutions of the Generalized (3 + 1)-Dimensional Kadomtsev-Petviashvilli Equation with Variable Coeffi- cients,” Chaos Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 217-226. doi:10.1016/j.chaos.2005.08.148
[16]	M. A. Abdou, “Further Improved F-Expansion and New Exact Solutions for Nonlinear Evolution Equations,” Non- linear Dynamics, Vol. 52, No. 3, 2008, pp. 277-288. doi:10.1007/s11071-007-9277-3
[17]	M. Wang, X. Li and J. Zhang, “The (G'/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evo- lution Equations in Mathematical Physics,” Physics Let- ters A, Vol. 372, No. 4, 2008, pp. 417-423. doi:10.1016/j.physleta.2007.07.051
[18]	J. Zhang, X. Wei and Y. J. Lu, “A Generalized (G'/G)- Expansion Method and Its Applications,” Physics Letters A, Vol. 372, No. , 2008, pp. 36-53. doi:10.1016/j.physleta.2008.01.057
[19]	A. Bekir, “Application of the (G'/G)-Expansion Method for Nonlinear Evolution Equations,” Physics Letters A, Vol. 372, No. 19, 2008, pp. 3400-3406.
[20]	A. Bekir and A.C. Cevikel, “New Exact Travelling Wave Solutions of Nonlinear Physical Models,” Chaos Solitons & Fractals, Vol. 41, No. 4, 2008, pp. 1733-1739.
[21]	E. M. E. Zayed and K. A. Gepreel, “Some Applications of the (G'/G)-Expansion Method to Non-Linear Partial Dif- ferential Equations,” Applied Mathematics and Computa- tion, Vol. 212, No. 1, 2009, pp. 1-13. doi:10.1016/j.amc.2009.02.009
[22]	D. D. Ganji and M. Abdollahzadeh, “Exact Traveling So- lutions of Some Nonlinear Evolution Equation by (G'/G)- Expansion Method,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, Article ID: 013519. doi:10.1063/1.3052847
[23]	M. Wang, J. Zhang and X. Li, “Application of the (G'/G)- Expansion to Travelling Wave Solutions of the Broerkaup and the Approximate Long Water Wave Equations,” Ap- plied Mathematics and Computation, Vol. 206, No. 1, 2008, pp. 321-326. doi:10.1016/j.amc.2008.08.045
[24]	L.-X. Li and M.-L.Wand, “The (G'/G)-Expansion Method and Travelling Wave Solutions for a Higher-Order Non- linear Schrdinger Equation,” Applied Mathematics and Computation, Vol. 208, No. 2, 2009, pp. 440-445. doi:10.1016/j.amc.2008.12.005
[25]	E. M. E. Zayed and K. A. Gepreel, “The (G'/G)-Expan- sion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics,” Journal of Mathematical Physics, Vol. 50, No. 1, 2008, Article ID: 013502. doi:10.1063/1.3033750
[26]	I. Aslan and T. Ozis, “Analytic Study on Two Nonlinear Evolution Equations by Using the (G'/G)-Expansion Me- thod,” Applied Mathematics and Computation, Vol. 209, No. 2, 2009, pp. 425-429. doi:10.1016/j.amc.2008.12.064
[27]	I. Aslan and T. Ozis, “On the Validity and Reliability of the (G'/G)-Expansion Method by Using Higher-Order Non- linear Equations,” Applied Mathematics and Computation, Vol. 211, No. 2, 2009, pp. 531-536. doi:10.1016/j.amc.2009.01.075