Classification of travelling waves in the Fornberg–Whitham equation

In this paper, the Fornberg–Whitham (FW) equation is investigated by using the improved qualitative method, which combines characteristics of several methods to solve equations. We classify all travelling wave solutions of this equation in specified regions of the parametric space. Besides already known solutions such as peaked and kink-like wave solutions, this equation also admits looped and cusped wave solutions, especially some composite waves including fractal-like waves and stumpons. What is more interesting is that we give the limit behavior of all periodic solutions as the parameters trend to some special values.