Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation. (English) Zbl 0997.35096

Summary: Bifurcations of solitary waves and kink waves for the generalized Pochhammer-Chree equation \[ u_{tt}-u_{ttxx}-\sigma(u)_{xx}=0, \] are studied, by using the bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Numbers of solitary waves and kink waves are given. Under various parameter conditions, all explicit formulas of solitary wave solutions and kink wave solutions are obtained.


35Q58 Other completely integrable PDE (MSC2000)
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35C05 Solutions to PDEs in closed form
Full Text: DOI


[1] Bogolubsky, I.L., Some examples of inelastic soliton interaction, Comput. phys. commun., 13, 2, 149-155, (1977)
[2] Clarcson, P.A.; LeVeque, R.J.; Saxton, R., Solitary wave interactions in elastic rods, Stud. appl. math., 75, 1, 95-122, (1986) · Zbl 0606.73028
[3] Chow, S.N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer New York
[4] Guckenheimer, J.; Holmes, P., Dynamical systems and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001
[5] Li, J.; Liu, Z., Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl. math. model., 25, 41-56, (2000) · Zbl 0985.37072
[6] Saxton, R., Existence of silutions for a finite nonlinearly hyperelestic rod, J. math. anal. appl., 105, 1, 59-75, (1985) · Zbl 0561.73046
[7] Zhang, W.; Ma, W., Explicit solitary wave solutions to generalized pochhammer – chree equations, Appl. math. mech., 20, 6, 666-674, (1999) · Zbl 0935.35132
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.