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.


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


Zbl 1089.65113
Full Text: DOI


[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
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