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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 156139, 14 pages
Research Article

Some Further Results on Traveling Wave Solutions for the ZK-BBM( ) Equations

1College of Mathematics and Information Sciences, Shaoguan University, Shaoguan, Guangdong 512005, China
2Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

Received 30 June 2013; Accepted 4 September 2013

Academic Editor: K. S. Govinder

Copyright © 2013 Shaoyong Li and Zhengrong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the traveling wave solutions for the ZK-BBM( ) equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.