Integral bifurcation method together with a translation-dilation transformation for solving an integrable 2-component Camassa-Holm shallow water system

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. © 2012 weiguo rui and yao long.