New Explicit Solutions of the Generalized (2 + 1)-Dimensional Zakharov-Kuznetsov Equation

Abstract

his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.

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G. Wang, X. Liu and Y. Zhang, "New Explicit Solutions of the Generalized (2 + 1)-Dimensional Zakharov-Kuznetsov Equation," Applied Mathematics, Vol. 3 No. 6, 2012, pp. 523-527. doi: 10.4236/am.2012.36079.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. J. Ablowitz and H. Segur, “Solitons and Inverse Scattering Transform,” SIAM, Philadelphia, 1981. doi:10.1137/1.9781611970883
[2] G. T. Liu and T. Y. Fan, “New Applications of Developed Jacobi Function Expansion Methods,” Physics Letters A, Vol. 345, No. 1-3, 2005, pp. 161-166. doi:10.1016/j.physleta.2005.07.034
[3] R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridgeg, 2004.
[4] A. M. Wazwaz, “Distinct Variants of the KdV Equations with Compact and Noncompact Structures,” Applied Mathematics and Computation, Vol. 150, No. 2, 2004, pp. 365-377. doi:10.1016/S0096-3003(03)00238-8
[5] Z. L. Yan and X. Q. Liu, “Symmetry and Similarity Solutions of Variable Coefficients Generalized ZakharovKuznetsov Equation,” Applied Mathematics and Computation, Vol. 180, No. 1, 2006, pp. 288-294. doi:10.1016/j.amc.2005.12.021
[6] Z. L. Yan, X. Q. Liu and L.Wang, “The Direct Symmetry Method and Its Application in Variable Coefficients Schrodinger Equation,” Applied Mathematics and Computation, Vol. 187, No. 1, 2007, pp. 701-707. doi:10.1016/j.amc.2006.08.084
[7] P. J. Olver, “Application of Lie Group to Differential Equation,” Springer-Verlag, New York, 1993. doi:10.1007/978-1-4612-4350-2
[8] J. H. He and X. H. Wu, “Exp-Function Method for Nonlinear Wave Equations,” Chaos, Solitons & Fractals, Vol. 30, No. 3, 2006, pp. 700-708. doi:10.1016/j.chaos.2006.03.020
[9] M. Wang, X. Li and J. Zhang, “The (G'/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics,” Physics Letters A, Vol. 372, No. 4, 2008, pp. 417-423. doi:10.1016/j.physleta.2007.07.051
[10] Z. L. Li, “Constructing of New Exact Solutions to the GKdVmKdV Equation with Any-Order Nonlinear Terms by (G'/G)-Expansion Method,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 1398-1403. doi:10.1016/j.amc.2009.05.034
[11] M. Wang, J. Zhang and X. Li, “Application of the (G'/G)-Expansion Method to Traveling Wave Solutions of the Broer-Kaup and the Approximate Long Water Wave Equations,” Applied Mathematics and Computation, Vol. 206, No. 1, 2008, pp. 321-326. doi:10.1016/j.amc.2008.08.045
[12] S. Zhang, L. Dong, J. Ba and Y. Sun, “The (G'/G)-Expansion Method for Nonlinear Differential-Difference Equations,” Physics Letters A, Vol. 373, No. 10, 2009, pp. 905-910. doi:10.1016/j.physleta.2009.01.018
[13] V. E. Zakharov and E. A. Kuznetsov, “On Three-Dimensional Solitons,” Soviet Physics, Vol. 39, 1974, pp. 285288.
[14] S. Munro and E. J. Parkes, “The Derivation of a Modified Zakharov-Kuznetsov Equation and the Stability of Its Solutions,” Journal of Plasma Physics, Vol. 62, No. 3, 1999, pp. 305-317. doi:10.1017/S0022377899007874
[15] J. Das, A. Bandyopadhyay and K. P. Das, “Stability of an Alternative Solitary-Wave Solution of an Ionacoustic Wave Obtained from the MKdV-KdV-ZK Equation in Magnetized Non-Thermal Plasma Consisting of Warm Adiabaticions,” Journal of Plasma Physics, Vol. 72, 2006, pp. 587-604. doi:10.1017/S0022377805004290
[16] A. Biswas and E. Zerrad, “1-Soliton Solution of the Zakharov-Kuznetsov Equation with Dual-Power Law Nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 9-10, 2009, pp. 35743577. doi:10.1016/j.cnsns.2008.10.004
[17] J. H. He, “Application of Homotopy Perturbation Method to Nonlinear Wave Equations,” Chaos, Solitons & Fractals, Vol. 26, No. 1, 2005, pp. 695-700. doi:10.1016/j.chaos.2005.03.006
[18] Z. Fu, S. Liu and S. Liu, “Multiple Structures of 2-D Nonlinear Rossby Wave,” Chaos, Solitons & Fractals, Vol. 24, 2005, pp. 383-390.
[19] I. Aslan, “Generalized Solitary and Periodic Wave Solutions to a (2 + 1)-Dimensional Zakharov-Kuznetsov Equation,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 1421-1429. doi:10.1016/j.amc.2009.05.037

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