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Bifurcations of travelling wave solutions for the Gilson-Pickering equation.(English)Zbl 1177.35161

Summary: The qualitative behavior and exact travelling wave solutions of the Gilson-Pickering equation are studied by using the qualitative theory of polynomial differential system. The phase portraits of the system are given under different parametric conditions. Some exact travelling wave solutions of the Gilson-Pickering equation are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B32 Bifurcations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B10 Periodic solutions to PDEs
35C05 Solutions to PDEs in closed form

Cite

References:

[1]Clarkson, P. A., Symmetries of a class of nonlinear third-order partial differential equations, Math. Comput. Modelling, 25, 195-212 (1997) ·Zbl 0879.35005
[2]Fornberg, B.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R Soc. Lond. A, 289, 373-404 (1978) ·Zbl 0384.65049
[3]Li, J.; Wu, J.; Zhu, H., Traveling waves for an integrable higher order KdV type wave equations, Internat. J. Bifur. Chaos, 16, 8, 2235-2260 (2006) ·Zbl 1192.37100
[4]Shen, J.; Xu, W., Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 26, 1149-1162 (2005) ·Zbl 1072.35579
[5]Li, J.; Liu, Z., Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl. Math. Modelling, 25, 41-56 (2000) ·Zbl 0985.37072
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[7]Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) ·Zbl 0972.35521
[8]Gilson, C.; Pickering, A., Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations, J. Phys. A: Math. Gen., 28, 2871-2888 (1995) ·Zbl 0830.35127
[9]Fuchssteiner, B.; Fokas, A. S., Symplectic structure, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981) ·Zbl 1194.37114
[10]Cairó, L.; Llibre, J., Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2, Nonlinear Anal., 67, 327-348 (2007) ·Zbl 1189.34065
[11]Li, J.; Dai, H. H., On the Study of Singular Nonlinear traveling wave equations, (Dynamical System Approach (2007), Science Press: Science Press Beijing), (English)
[12]Li, J.; Chen, G., On a class of singular nonlinear traveling wave equations, Internat. J. Bifur. Chaos, 17, 4049-4065 (2007) ·Zbl 1158.35080
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.
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