The integral bifurcation method and its application for solving a family of third-order dispersive PDEs

In this paper, an improved method named the integral bifurcation method is introduced. In order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations, a family of third-order dispersive partial differential equations which were given by A. Degasperis, D. Holm and A. Hone are studied. Many integral bifurcations are obtained for different parameter conditions. By using these integral bifurcations, many travelling wave solutions such as loop soliton solutions, solitary wave solutions, cusp soliton solutions and periodic wave solutions are obtained. In particular, under the conditions c1<0,c2=c3=1c1<0,c2=c3=1, a very peculiar periodic wave solution is obtained.