×

zbMATH — the first resource for mathematics

Bifurcations of travelling wave solutions for the \(K(n,-n,2n)\) equations. (English) Zbl 1157.37021
For the following \(K(n,-n,2n)\) equations \(u_t=a(u'')_x+b[u^{-n}(u^{2n})_{xx}]_{x}=0\) where \(a,b\) are two nonzero real numbers, \(n\) is positive integer. A. M. Wazwaz [Appl. Math. Comput., 173, No. 1, 213–230 (2006; Zbl 1089.65113)] has considered some compact and non-compact solutions of travelling wave structure. The authors investigate bifurcations of travelling wave solutions in the five-parameter space \((a, b, c, g, n)\) and prove the existence of solitary wave solutions, solitary cusp wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions.

MSC:
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
35B32 Bifurcations in context of PDEs
35Q51 Soliton equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
Citations:
Zbl 1089.65113
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Wazwaz, A.M., Explicit travelling wave solutions of variants of the \(K(n, n)\) and \(\mathit{ZK}(n, n)\) equations with compact and noncompact structures, Appl. math. comput., 173, 213-230, (2006) · Zbl 1089.65113
[2] Chow, S.N.; Hale, J.K., Method of bifurcation theory, (1981), Springer-Verlag New York
[3] Li, Jibin; Liu, Zhenrong, Smooth and non-smooth travelling waves in a non-linearly dispersive equation, Appl. math. model., 25, 41-56, (2000) · Zbl 0985.37072
[4] Li, Jibin; Liu, Zhenrong, Travelling wave solutions for a class of nonlinear dispersive equations, Chin. ann. math., 23B, 397-418, (2002) · Zbl 1011.35014
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.