Global dynamical behaviors in a physical shallow water system

The theory of bifurcations of dynamical systems is used to investigate the behavior of travelling wave solutions in an entire family of shallow water wave equations. This family is obtained by a perturbative asymptotic expansion for unidirectional shallow water waves. According to the parameters of the system, this family can lead to different sets of known equations such as Camassa-Holm, Korteweg-de Vries, Degasperis and Procesi and several other dispersive equations of the third order. Looking for possible travelling wave solutions, we show that different phase orbits in some regions of parametric planes are similar to those obtained with the model of the pressure waves studied by Li and Chen. Many other exact explicit travelling waves solutions are derived as well, some of them being in perfect agreement with solutions obtained in previous works by researchers using different methods. When parameters are varied, the conditions under which the above solutions appear are also shown. The dynamics of singular nonlinear travelling system is completely determined for each of the above mentioned equations. Moreover, we define sufficient conditions leading to the existence of propagating wave solutions and demonstrate how and why travelling waves lose their smoothness and develop into solutions with compact support or breaking waves. (C) 2015 Elsevier B.V. All rights reserved.