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$R(m, n)$ equations; solitary wave; periodic wave; breaking wave; solitary cusp wave; periodic cusp wave.

By using the bifurcation theory and methods of planar dynamical systems to $R(m, n)$ equations, the dynamical behavior of different physical structures like smooth and non-smooth solitary wave, kink wave, smooth and non-smooth periodic wave, and breaking wave is obtained. The qualitative change in the physical structures of these waves is shown to depend on the systemic parameters. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above waves are given. Moreover, some explicit exact parametric representations of travelling wave solutions are listed.

Dahe Feng¹ Jibin Li²

1. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, People’s Republic of China
2. Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of China

Published

November 2007

Manuscript received

17 September 2006

Manuscript revised

14 January 2007