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The integral bifurcation method and its application for solving a family of third-order dispersive PDEs. (English) Zbl 1144.35461

Summary: 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 \(c_{1}<0,c_{2}=c_{3}=1\), a very peculiar periodic wave solution is obtained.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35B10 Periodic solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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