Bifurcations of travelling wave solutions for the generalized \((2+1)\)-dimensional Boussinesq-Kadomtsev-Petviashvili equation. (English) Zbl 1202.35270
Summary: By using the bifurcation theory of dynamical systems, we study the generalized \((2+1)\)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, the existence of solitary wave solutions, compacton solutions, periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35C07 | Traveling wave solutions |
| 35B32 | Bifurcations in context of PDEs |
| 37K50 | Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35C08 | Soliton solutions |
| 35B10 | Periodic solutions to PDEs |
Keywords:
solitary wave; periodic cusp wave; compacton solutions; periodic travelling wave solution; generalized \((2+1)\)-dimensional Boussinesq-Kadomtsev-Petviashvili equationReferences:
| [1] | Tascan, F.; Bekir, A., Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method, Appl. Math. Comput., 215, 3134-3139 (2009) · Zbl 1180.35154 |
| [2] | Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP type, Physica D, 4, 343-365 (1982) · Zbl 0571.35100 |
| [3] | Ma, H. C.; Wang, Y.; Qin, Z. Y., New exact complex traveling wave solutions for (2+1)-dimensional BKP equation, Appl. Math. Comput., 208, 564-568 (2009) · Zbl 1168.35425 |
| [4] | Jeffry, A.; Kakutani, T., Weak nonlinear dispersive waves, a discussion centered around the Korteweg-de Vries equation, SIAM Rev., 14, 582-643 (1972) · Zbl 0221.35038 |
| [5] | Manakov, S. V.; Zakharov, V. E.; Bordag, L. A.; Its, A. R.; Matveev, V. B., Two dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett., 63A, 205-206 (1977) |
| [6] | Johan, V. D.L., The \(n\)-th reduced BKP hierarchy, the string equation and \(BW_1\)+∞ constraints, Acta Appl. Math., 44, 185-206 (1996) · Zbl 0866.17017 |
| [7] | Chow, Shui-Nee; Hale, J. K., Method of Bifurcation Theory (1981), Springer-Verlag: Springer-Verlag New York |
| [8] | Guckenheimer, J.; Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0515.34001 |
| [9] | Perko, L., Differential Equations and Dynamical Systems (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0717.34001 |
| [10] | Tang, S.; Huang, W., Bifurcations of travelling wave solutions for the \(K(n\),−\(n,2n)\) equations, Appl. Math. Comput., 203, 39-49 (2008) · Zbl 1157.37021 |
| [11] | Li, J. B.; Liu, Z., Travelling wave solutions for a class of nonlinear dispersive equations, Chin. Ann. Math., 23B, 397-418 (2002) · Zbl 1011.35014 |
| [12] | Tang, S.; Huang, W., Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation, Appl. Math.Comput., 189, 1774-1781 (2007) · Zbl 1124.35077 |
| [13] | Byrd, PF.; Fridman, MD., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Springer: Springer Berlin · Zbl 0213.16602 |
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.
