×

Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation. (English) Zbl 1045.37045

Summary: By using the theory of bifurcations of dynamical systems to the generalization form of the modified KdV equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and nonsmooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

MSC:

37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53 KdV equations (Korteweg-de Vries equations)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chow, S.N.; Hale, J.K., Method of bifurcation theory, (1981), Springer-Verlag New York
[2] Debnath, L., Nonlinear partial differential equations for scientists and engineers, (1997), Birkhauser Boston · Zbl 0892.35001
[3] Fokas, A.S., On class of physically important integrable equations, Physica D, 87, 145-150, (1995) · Zbl 1194.35363
[4] Guckenheimer, J.; Holmes, P.J., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[5] Li, J.; Liu, Z., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl. math. modelling, 25, 41-56, (2000) · Zbl 0985.37072
[6] Li, J.; Liu, Z., Travelling wave solutions for a class of nonlinear dispersive equations, Chinese ann. math. ser. B, 23, 3, 397-418, (2002) · Zbl 1011.35014
[7] Li, Y.A.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. I. compactons and peakons, Discr. cont. dynam. syst., 3, 419-432, (1997) · Zbl 0949.35118
[8] Li, Y.A.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. II. complex analytic behaviour and convergence to non-analytic solutions, Discr. cont. dynam. syst., 4, 159-191, (1998) · Zbl 0959.35157
[9] Perko, L., Differential equations and dynamical systems, (1991), Springer-Verlag New York · Zbl 0717.34001
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.