×

zbMATH — the first resource for mathematics

The application of bifurcation method to a higher-order KdV equation. (English) Zbl 1012.35076
Summary: Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the higher-order KdV equation \[ u_t+au^nu_x +u_{xxx}=0, \] where \(n\geq 1\) and \(a\in\mathbb{R}\). Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, explicit solitary wave solutions are obtained. Specially, some new solitary wave solutions are found for the KdV or MKDV equation.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Miura, R.M., The korteweg – de Vries equation: a survey of results, SIAM rev., 8, 412-459, (1976) · Zbl 0333.35021
[2] Dodd, R.K.; Eilbeck, J.C.; Gibbon, J.D.; Morris, H.C., Solitons and nonlinear wave equations, (1982), Academic Press London · Zbl 0496.35001
[3] Bona, J.L.; Dougalis, V.A.; Karakashian, O.A.; Mckinney, W.R., Conservative, high-order numerical schemes for the generalized korteweg – de Vries equation, Philos. trans. roy. soc. London ser. A, 351, 107-164, (1995) · Zbl 0824.65095
[4] Fornberg, B.; Whitham, G.B., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. trans. roy. soc. London ser. A, 289, 373-404, (1978) · Zbl 0384.65049
[5] Chow, S.N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer-Verlag New York
[6] Guckenheimer, J.; Holmes, P., Dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[7] Dey, B., K-dv like equations with domain wall solutions and their Hamiltonians, (), 188-194
[8] Dey, B., Domain wall solutions of KdV like equations with higher order nonlinearity, J. phys. A, 19, L9-L12, (1986) · Zbl 0624.35070
[9] Lamb, G.L., Elements of soliton theory, (1980), John Wiley and Sons New York · Zbl 0445.35001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.