By applying the bifurcation theory of dynamical system to the generalized KP-BBM equation, the phase portraits of the travelling wave system are obtained. It can be shown that singular straight line in the travelling wave system is the reason why smooth periodic waves converge to periodic cusp waves. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.

The bifurcation theory of dynamical systems is applied to an integrable non-linear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.

It is showed that the fully nonlinear evolution equations of Olver and Rosenau can be reduced to Hamiltonian form by transformation of variables. The resulting Hamiltonian equations are treated by the dynamical systems theory and a phase-space analysis of their singular points is presented. The results of this study demonstrate that the equations can support double compactons. The new Olver–Rosenau compactons are different from the well-known Rosenau–Hyman compacton and Cooper–Shepard–Sodano compacton, because they are induced by a singular elliptic instead of singular straight line on phase-space.

The periods of the periodic travelling wave solutions for an elastic beam equation were studied. By the transformation of variables, the elastic beam equation was reduced to a planar differential system. The period function of the planar differential system is examined. Itwas proved that the period function is a monotonic function.Moreover, the asymptotic behaviour of the period function was also given.