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The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations. (English) Zbl 1152.35485

Summary: We establish exact solutions for nonlinear evolution equations. The tanh and sine-cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations. Many new families of exact travelling wave solutions of the Vakhnenko and modified Benjamin-Bona-Mahony (MBBM) equations are successfully obtained. The obtained solutions include solitons, solitary and periodic solutions. These solutions may be important of significance for the explanation of some practical physical problems.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B10 Periodic solutions to PDEs
35Q51 Soliton equations
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[1] Ablowitz, M. J.; Segur, H., Solitons and inverse scattering transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076
[2] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos Trans Roy Soc London Ser A, 272, 47-48 (1972) · Zbl 0229.35013
[3] El-Wakil, S. A.; Abdou, M. A., New exact travelling wave solutions using modified extended tanh-function method, Chaos, Solitons & Fractals, 31, 4, 840-852 (2007) · Zbl 1139.35388
[4] El-Wakil, S. A.; Abdou, M. A., Modified extended tanh-function method for solving nonlinear partial differential equations, Chaos, Solitons & Fractals, 31, 5, 1256-1264 (2007) · Zbl 1139.35389
[5] Fan, E.; Zhang, H., A note on the homogeneous balance method, Phys Lett A, 246, 403-406 (1998) · Zbl 1125.35308
[6] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys Lett A, 277, 212 (2000) · Zbl 1167.35331
[7] Fan, E.; Hon, Y. C., Generalized tanh method extended to special types of nonlinear equations, Z Naturforsch, 57a, 692-700 (2002)
[8] Hirota, R., Direct method of finding exact solutions of nonlinear evolution equations, (Bullough, R.; Caudrey, P., Backlund transformations (1980), Springer: Springer Berlin), 1157-1175
[9] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am J Phys, 60, 650-654 (1992) · Zbl 1219.35246
[10] Khater, A. H.; Malfliet, W.; Callebaut, D. K.; Kamel, E. S., The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations, Chaos, Solitons & Fractals, 14, 2, 513-522 (2002) · Zbl 1002.35066
[11] Tso, T., Existence of solutions of the modified Benjamin-Bone-Mahony equation, Chin J Math, 24, 4, 327-336 (1996) · Zbl 0867.35021
[12] Vakhnenko, V. O.; Parkes, E. J., The two loop soliton of the Vakhnenko equation, Nonlinearity, 11, 1457-1464 (1998) · Zbl 0914.35115
[13] Vakhnenko, V. O.; Parkes, E. J., The calculation of multi-soliton solutions of the Vakhnenko equation, Chaos, Soliton & Fractals, 13, 1819-1826 (2002) · Zbl 1067.37106
[14] Vakhnenko, V. O.; Parkes, E. J.; Morrison, A. J., A B äcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation, Chaos, Solitons & Fractals, 17, 4, 683-692 (2003) · Zbl 1030.37047
[15] Yan, Z., Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation II: solitary pattern solutions, Chaos, Solitons & Fractals, 18, 4, 869-880 (2003) · Zbl 1068.35148
[16] Yan, Z., Abundant families of Jacobi elliptic function solutions of the \((2 + 1)\)-dimensional integrable Davey-Stewartson-type equation via a new method, Chaos, Solitons & Fractals, 18, 2, 299-309 (2003) · Zbl 1069.37060
[17] Yusufoglu, E.; Bekir, A., Exact Solutions of Coupled Nonlinear Evolution Equations, Chaos, Solitons & Fractals, 37, 3, 842-848 (2008) · Zbl 1148.35352
[18] Yusufoglu, E.; Bekir, A.; Alp, M., Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine-Cosine method, Chaos, Solitons & Fractals, 37, 4, 1193-1197 (2008) · Zbl 1148.35351
[19] Wadati, M., Introduction to solitons, Pramana: J Phys, 57, 5-6, 841-847 (2001)
[20] Wadati, M., The exact solution of the modified Korteweg-de Vries equation, J Phys Soc Jpn, 32, 1681-1687 (1972)
[21] Wadati, M., The modified Korteweg-de Vries equation, J Phys Soc Jpn, 34, 1289-1296 (1973) · Zbl 1334.35299
[22] Wazwaz, A. M., Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos, Solitons & Fractals, 28, 2, 454-462 (2006) · Zbl 1084.35079
[23] Wazwaz, A. M., The tanh method: exact solutions of the Sine-Gordon and Sinh-Gordon equations, Appl Math Comput, 167, 1196-1210 (2005) · Zbl 1082.65585
[24] Wazwaz, A. M., The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos, Solitons & Fractals, 25, 1, 55-63 (2005) · Zbl 1070.35076
[25] Wazwaz, A. M., The sine-cosine method for obtaining solutions with compact and noncompact structures, Appl Math Comput, 159, 2, 559-576 (2004) · Zbl 1061.35121
[26] Wazwaz, A. M.; Helal, M. A., Nonlinear variants of the BBM equation with compact and noncompact physical structures, Chaos, Solitons & Fractals, 26, 3, 767-776 (2005) · Zbl 1078.35110
[27] Wazwaz, A. M., New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, Solitons & Fractals, 22, 1, 249-260 (2004) · Zbl 1062.35121
[28] Wazwaz, A. M., A sine-cosine method for handling nonlinear wave equations, Math Comput Model, 40, 499-508 (2004) · Zbl 1112.35352
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