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Mathematical Problems in Engineering
Volume 2011, Article ID 810217, 23 pages
http://dx.doi.org/10.1155/2011/810217
Research Article

Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 15 February 2011; Accepted 26 May 2011

Academic Editor: Ming Li

Copyright ยฉ 2011 Liu Zhengrong et al. 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.

Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation ๐‘ข ๐‘ก + ๐‘Ž ( 1 + ๐‘ ๐‘ข ) ๐‘ข ๐‘ข ๐‘ฅ + ๐‘ข ๐‘ฅ ๐‘ฅ ๐‘ฅ = 0 , and PC-like equation ๐‘ฃ ๐‘ก ๐‘ก โˆ’ ๐‘ฃ ๐‘ก ๐‘ก ๐‘ฅ ๐‘ฅ โˆ’ ( ๐‘Ž 1 ๐‘ฃ + ๐‘Ž 2 ๐‘ฃ 2 + ๐‘Ž 3 ๐‘ฃ 3 ) ๐‘ฅ ๐‘ฅ = 0 , respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions. Our work extend previous some results.