Bifurcations of traveling wave solutions for a class of the generalized Benjamin–Bona–Mahony equation

By using the theory of bifurcations of dynamical systems to a class of the generalized Benjamin–Bona–Mahony (GBBM) equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular curves in a traveling wave system is the reason why smooth waves converge to cusp waves, finally. When parameters are varied, under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.