New exact traveling wave solutions of some nonlinear higher-dimensional physical models. (English) Zbl 1284.35374

Summary: In this paper, some new traveling wave solutions of the \((4+1)\)-dimensional Fokas equation, \((3+1)\)-dimensional Jumbo-Miwa equation and \((2+1)\)-dimensional Boiti-Leon-Pempinelli equation are obtained through the \((\frac{G'}{G})\)-expansion technique. The key idea of this technique is to take full advantage of a Riccati equation involving two parameters and use its solutions in obtaining the traveling wave solutions. The results reveal that this technique is very effective and powerful for solving higher-dimensional nonlinear problems arising in mathematical physics.


35Q53 KdV equations (Korteweg-de Vries equations)
35C07 Traveling wave solutions
68W30 Symbolic computation and algebraic computation


ATFM; Maple
Full Text: DOI Link


[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering Transformation (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0762.35001
[2] Bekir, A., Application of the \((\frac{G^\prime}{G})\)-expansion method for nonlinear evolution equations, Phys. Lett. A, 372, 3400-3406 (2008) · Zbl 1228.35195
[3] Bekir, A.; Cevikel, A. C., New exact traveling wave solutions of nonlinear physical models, Chaos Solitons Fractals, 41, 1733-1739 (2009) · Zbl 1198.35049
[4] Dai, C. Q.; Wang, Y. Y., New exact solutions of the (3 + 1)-dimensional Burgers system, Phys. Lett. A, 373, 181-187 (2009) · Zbl 1227.35231
[5] Dai, C. Q.; Cen, X.; Wu, S. S., Exact traveling wave solutions of the discrete sine Gordon equation obtained via the exp-function method, Nonlinear Analysis, 70, 58-63 (2009) · Zbl 1183.34101
[6] Dai, C. Q.; Y Wang, Y., New variable separation solutions of the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov system, Nonlinear Analysis, 71, 1496-1503 (2009) · Zbl 1173.35315
[7] Mohyud-Din, S. T.; Noor, M. A.; Noor, K. I., Variational iteration method for re-formulated partial differential equations, Internat. J. Nonlinear Sci., 11, 87-92 (2010) · Zbl 1401.65141
[8] Hesameddini, E.; Latifizadeh, H., Reconstruction of variational iteration algorithms using the Laplace transform, Internat. J. Nonlinear Sci., 10, 1377-1382 (2009)
[9] Dehghan, M.; Shakeri, F., Use of He’s homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, Journal Porous Media, 11, 765-778 (2008)
[10] Dehghan, M.; Manafian, J., The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method, Zeitschrift Naturforschung A, 64a, 411-419 (2009)
[11] Dehghan, M.; Shakeri, F., Application of He’s variational iteration method for solving the Cauchy reaction-diffusion problem, J. Comput. Appl. Math., 214, 435-446 (2008) · Zbl 1135.65381
[12] El-Wakil, SA; Abdou, MA, New exact traveling wave solutions using modified extended tanh-function method, Chaos Solitons Fract., 31, 840-852 (2007) · Zbl 1139.35388
[13] Fan, E., Extended tanh-function method, its applications to nonlinear equations, Phys. Lett. A, 277, 212-218 (2000) · Zbl 1167.35331
[14] Fokas, A. S., A symmetry approach to exactly solvable evolution equations, J. Math. Phys., 21, 1318-1325 (1980) · Zbl 0455.35109
[15] Hu, J. L., A new method for finding exact traveling wave solutions to nonlinear partial differential equations, Phys. Lett. A, 286, 175-179 (2001) · Zbl 0969.35532
[16] Hu, J. L., A new method of exact traveling wave solution for coupled nonlinear differential equations, Phys. Lett. A, 322, 211-216 (2004) · Zbl 1118.81366
[17] He, J. H.; Abdou, M. A., New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Solitons Fractals, 34, 1421-1429 (2007) · Zbl 1152.35441
[18] He, J. H.; Zhang, L. N., Generalized solitary solution, compacton-like solution of the Jaulent-Miodek equations using the Exp-function method, Phys. Lett. A, 372, 1044-1047 (2008) · Zbl 1217.35152
[19] Jimbo, M.; Miwa, T., Solitons, infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19, 943-1001 (1983) · Zbl 0557.35091
[20] Luwai, Wazzan, A modified tanh-coth method for solving the general Burgers-Fisher and the Kuramoto-Sivashinsky equations, Commun. Nonlinear Sci. Numer. Simul., 14, 2642-2652 (2009) · Zbl 1221.35320
[21] Luwai, Wazzan, A modified tanh-coth method for solving the KdV and the KdV-Burgers’ equations, Commun. Nonlinear Sci. Numer. Simul., 14, 443-450 (2009) · Zbl 1221.35376
[22] Li, X. Z.; Wang, M. L., A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms, Phys. Lett. A, 361, 115-118 (2007) · Zbl 1170.35085
[23] Li, B.; Chen, Y.; Xuan, H.; Zhang, H., Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo-Miwa equation, Appl. Math. Comput., 152, 581-595 (2004) · Zbl 1049.65106
[24] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun., 98, 288-300 (1998) · Zbl 0948.76595
[25] Sakthivel, R.; Chun, C., New soliton solutions of Chaffee-Infante equations, Z. Naturforsch. A (J. Phys. Sci.), 65a, 197-202 (2010)
[26] Sakthivel, R.; Chun, C.; Lee, J., New traveling wave solutions of Burgers equation with finite transport memory, Z. Naturforsch. A (J. Phys. Sci.), 65, 633-640 (2010)
[27] Chun, C.; Sakthivel, R., Homotopy perturbation technique for solving two-point boundary value problems comparison with other methods, Computer Physics Communications, 181, 1021-1024 (2010) · Zbl 1216.65094
[28] Chun, C.; Sakthivel, R., New soliton, periodic solutions for two nonlinear physical models, Z. Naturforsch. A (J. Phys. Sci.), 65a, 1049-1054 (2010)
[29] Kim, H.; Sakthivel, R., Travelling wave solutions for time-delayed nonlinear evolution equations, Applied Mathematics Letters, 23, 527-532 (2010) · Zbl 1189.35281
[30] Kudryashov, N. A., Seven common errors in finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 14, 3507-3539 (2009) · Zbl 1221.35342
[31] Senthilvelan, M., On extended applications of homogeneous balance method, Appl. Math. Comput., 123, 381-388 (2001) · Zbl 1032.35159
[32] Wang, M. L.; Li, X.; Zhang, J., The \((\frac{G^\prime}{G})\)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372, 417-423 (2008)
[33] Wang, M. L.; Li, X. Z.; Zhang, J. L., Sub-ODE method and solitary wave solutions for higher order nonlinear Schröodinger equation, Phys. Lett. A, 363, 96-101 (2007) · Zbl 1197.81129
[34] Zheng, Y. Z.; Ya, Y. Z., Symmetry groups, exact solutions of new (4 + 1)-dimensional Fokas equtaion, Commun. Theor. Phys, 51, 876-880 (2009) · Zbl 1177.35210
[35] Zhang, S.; Tong, J. L.; Wang, W., A generalized \((\frac{G^\prime}{G})\)-expansion method for the mKdV equation with variable coefficients, Phys. Lett. A, 372, 2254-2257 (2008) · Zbl 1220.37072
[36] Zhang, S.; Wang, W.; Tong, J. L., A generalized \((\frac{G^\prime}{G})\)-expansion method and its application to the (2 + 1)-dimensional Broer-Kaup equations, Appl. Math. Comput., 209, 399-404 (2009) · Zbl 1165.35457
[37] Boiti, M.; Leon, J. J.P.; Pempinelli, F., Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations, Inverse. Prob., 3, 37-49 (1987) · Zbl 0625.35073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.