Bifurcations of travelling wave solutions for the \(K(n,-n,2n)\) equations. (English) Zbl 1157.37021

Tang, Shengqiang; Huang, Wentao

Appl. Math. Comput. 203, No. 1, 39-49 (2008).

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.

Reviewer: Boris V. Loginov (Ul’yanovsk)

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

Keywords:

solitary wave solution; periodic cusp wave solutions; kink and anti-kink wave solution; periodic travelling wave solution; the \(K(n,-n,2n)\) equations

Citations:

Zbl 1089.65113

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