Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation

In this paper, we study the nonlinear wave solutions of the generalized Camassa-Holm-Degasperis-Procesi equation $ u_t-u_{x x t}+(1+b)u^2u_x=b u_x u_{x x}+u u_{x x x}$. Through phase analysis, several new types of the explicit nonlinear wave solutions are constructed. Our concrete results are: (i) For given $b> -1$, if the wave speed equals $\frac{1}{1+b}$, then the explicit expressions of the smooth solitary wave solution and the singular wave solution are given. (ii) For given $b> -1$, if the wave speed equals $1+b$, then the explicit expressions of the peakon wave solution and the singular wave solution are got. (iii) For given $b> -2$ and $b\ne -1$, if the wave speed equals $\frac{2+b}{2}$, then the explicit smooth solitary wave solution, the peakon wave solution and the singular wave solution are obtained. We also verify the correctness of these solutions by using the software Mathematica. Our work extends some previous results.