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Solitary wave solutions and kink wave solutions for a generalized KdV-mKdV equation. (English) Zbl 1210.35214

Summary: Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized KDV-mKDV equation. Under some parameter conditions, their explicit expressions are obtained.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B32 Bifurcations in context of PDEs
35C08 Soliton solutions
35C05 Solutions to PDEs in closed form
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] Li, Z. L., Constructing of new exact solutions to the GKdV-mKdV equation with any-order nonlinear terms by (G′/G)-expansion method, Appl. Math. Comput., 217, 1398-1403 (2010) · Zbl 1203.35231
[2] 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
[3] Antonova, M.; Biswas, A., Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simul., 14, 734-748 (2009) · Zbl 1221.35321
[4] Smyth, N. F.; Worthy, A. L., Solitary wave evolution for mKdV equations, Wave Motion, 21, 263-275 (1995) · Zbl 0968.35517
[5] Khater, A. H., Two new classes of exact solutions for the KdV equation via Bäcklund transformations, Chaos Soliton Fract., 8, 12, 1901-1909 (1997) · Zbl 0938.35169
[6] Kovalyov, M.; Abadi, M. H.A., An explicit formula for a class of solutions of the KdV equation, Phys. Lett. A, 254, 47-52 (1999) · Zbl 0938.35162
[7] Gardner, L. R.T., Solitary wave solutions of the MKdV-equation, Comput. Method Appl. M., 124, 21-333 (1995) · Zbl 0945.65520
[8] Kevrekidis1, P. G., On some classes of mKdV periodic solutions, J. Phys. A: Math. Gen., 37, 10959-10965 (2004) · Zbl 1084.35073
[9] Wu, H. X., On the extended KdV equation with self-consistent sources, Phys. Lett. A, 370, 477-484 (2007) · Zbl 1209.35121
[10] Biswas, A.; Zerrad, E., Soliton perturbation theory for the Gardner equation, Adv. Stud. Theor. Phys., 2, 16, 787-794 (2008) · Zbl 1157.35466
[11] Zhang, J. F., Simple soliton solution method for the combined KdV and MKdV equation, Int. J. Theor. Phys., 39, 1697-1702 (2000) · Zbl 0958.35124
[12] Miura, R. M., The Korteweg-de Vries equation: a survey of results, SIAM Rev., 8, 412-459 (1976) · Zbl 0333.35021
[13] Liu, Z. R.; Yang, C. X., The application of bifurcation method to a higher-order KDV equation, J. Math. Anal. Appl., 275, 1, 1-12 (2002) · Zbl 1012.35076
[14] Li, J. B.; Liu, Z. R., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl. Math. Model., 25, 1, 41-56 (2000) · Zbl 0985.37072
[15] Li, J. B.; Zhang, L. J., Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos Soliton Fract., 14, 581-593 (2002) · Zbl 0997.35096
[16] Tang, M. Y.; Zhang, W. L., Four types of bounded wave solutions of CH-\(γ\) equation, Sci. Chin. Ser. A: Math., 50, 1, 132-152 (2007) · Zbl 1117.35310
[17] Song, M., Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 217, 1334-1339 (2010) · Zbl 1203.35245
[18] Song, M.; Cai, J. H., Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation, Appl. Math. Comput., 217, 1455-1462 (2010) · Zbl 1203.35244
[19] Chow, S. N.; Hale, J. K., Method of Bifurcation Theory (1982), Springer-verlag: Springer-verlag New York
[20] Guckenheimer, J.; Homes, P., Nonlinear Oscillations. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1999), Springer-verlag: Springer-verlag New York
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