New exact solutions for a higher-order wave equation of KdV type using the multiple simplest equation method. (English) Zbl 1442.35411

Summary: In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the simplest equation method and its variants. The solutions obtained are general solutions which are in the form of hyperbolic, trigonometric, and rational functions. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time.


35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
35C07 Traveling wave solutions
Full Text: DOI


[1] Wang, M., Exact solutions for a compound KdV-Burgers equation, Physics Letters. A, 213, 5-6, 279-287 (1996) · Zbl 0972.35526
[2] Yomba, E., The extended Fan’s sub-equation method and its application to KdV-{MK}dV, {BKK} and variant Boussinesq equations, Physics Letters A, 336, 6, 463-476 (2005) · Zbl 1136.35451
[3] Sirendaoreji; Jiong, S., Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309, 5-6, 387-396 (2003) · Zbl 1011.35035
[4] Yan, Z.; Zhang, H., New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Physics Letters A, 252, 6, 291-296 (1999) · Zbl 0938.35130
[5] Fan, E.; Zhang, J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters. A, 305, 6, 383-392 (2002) · Zbl 1005.35063
[6] Wu, X.; He, J., E{XP}-function method and its application to nonlinear equations, Chaos, Solitons & Fractals, 38, 3, 903-910 (2008) · Zbl 1153.35384
[7] Abdusalam, H. A., On an improved complex tanh-function method, International Journal of Nonlinear Sciences and Numerical Simulation, 6, 2, 99-106 (2005) · Zbl 1401.35012
[8] Wang, M.; Li, X.; Zhang, J., The G′/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters. A, 372, 4, 417-423 (2008) · Zbl 1217.76023
[9] Zayed, E. M. E.; Hoda Ibrahim, S. A., The two variable (G′/G, 1/G)-expansion method for finding exact traveling wave solutions of the (3+1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation, Proceedings of the International Conference on Advanced Computer Science and Electronics Information
[10] Zayed, E. M. E.; Hoda Ibrahim, S. A., The (G′/G, 1/G)-expansion method and its applications for constructing the exact solutions of the nonlinear Zoomeron equation, Transaction on IoT and Cloud Computing, 2, 1, 66-75 (2014)
[11] Kudryashov, N. A., Exact solitary waves of the Fisher equation, Physics Letters A, 342, 1-2, 99-106 (2005) · Zbl 1222.35054
[12] Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons and Fractals, 24, 5, 1217-1231 (2005) · Zbl 1069.35018
[13] Vitanov, N. K.; Dimitrova, Z. I., Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model {PDE}s from ecology and population dynamics, Communications in Nonlinear Science and Numerical Simulation, 15, 10, 2836-2845 (2010) · Zbl 1222.35201
[14] Bilige, S.; Chaolu, T., An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Applied Mathematics and Computation, 216, 11, 3146-3153 (2010) · Zbl 1195.35257
[15] Bilige, S.; Chaolu, T.; Wang, X., Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation, Applied Mathematics and Computation, 224, 517-523 (2013) · Zbl 1334.35308
[16] Fokas, A. S., On a class of physically important integrable equations, Physica D: Nonlinear Phenomena, 87, 1-4, 145-150 (1995) · Zbl 1194.35363
[17] Tzirtzilakis, E.; Marinakis, V.; Apokis, C.; Bountis, T., Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type, Journal of Mathematical Physics, 43, 12, 6151-6165 (2002) · Zbl 1060.35127
[18] Tzirtzilakis, E.; Xenos, M.; Marinakis, V.; Bountis, T. C., Interactions and stability of solitary waves in shallow water, Chaos, Solitons & Fractals, 14, 1, 87-95 (2002) · Zbl 1068.76011
[19] Li, J.; Rui, W.; Long, Y.; He, B., Travelling wave solutions for higher-order wave equations of KdV type (III), Mathematical Biosciences and Engineering, 3, 1, 125-135 (2006) · Zbl 1136.35449
[20] Rui, W. G.; Long, Y.; He, B., Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III), Nonlinear Analysis: Theory, Methods & Applications, 70, 11, 3816-3828 (2009) · Zbl 1167.34006
[21] Wu, X.; Rui, W.; Hong, X., A generalized KdV equation of neglecting the highest-order infinitesimal term and its exact traveling wave solutions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.35316
[22] He, Y.; Zhao, Y.; Long, Y., New exact solutions for a higher-order wave equation of KdV type using extended F-expansion method, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1296.35012
[23] Naher, H.; Abdullah, F. A., New traveling wave solutions by the extended generalized Riccati equation mapping method of the (2+1)-dimensional evolution equation, Journal of Applied Mathematics, 2012 (2012) · Zbl 1267.35067
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