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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 736765, 21 pages
Research Article

Integral Bifurcation Method together with a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm Shallow Water System

Center for Nonlinear Science Research, College of Mathematics, Honghe University, Yunnan, Mengzi 661100, China

Received 12 September 2012; Accepted 15 November 2012

Academic Editor: Michael Meylan

Copyright © 2012 Weiguo Rui and Yao Long. 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.


An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.