A Class Of Generalized Ch Equation, Traveling Wave Solutions

By using the bifurcation theory of dynamical systems to a class of the Generalized Camassa-Holm equation , the existence of smooth and non-smooth solitary wave solutions and uncountably infinite many smooth periodic wave solutions and non-smooth periodic wave solutions is shown. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.The paper consists of five parts. In the first part, there is some basic knowledge we need to know. In the second part, we introduce the corresponding model and results. In the third part, we discuss the bifurcations of phase portraits of (2.3) when f(u) = αu~2+βu~3. In the forth part, we shall give all explicit exact parametric representations for non-smooth solitary wave solutions and non-smooth periodic wave solutions of (2.1). Under some special parameter conditions, explicit exact parametric representations of some smooth solitary wave solutions are also given. In the fifth part, we show the existence of smooth solitary wave solutions smooth periodic wave solutions of (2.1).