Title: Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation
Abstract: In this paper, we study the nonlinear wave solutions ofthe 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 phaseanalysis, several new types of the explicit nonlinear wavesolutions are constructed. Our concrete results are: (i) For given$b> -1$, if the wave speed equals $\frac{1}{1+b}$, then theexplicit expressions of the smooth solitary wave solution and thesingular wave solution are given. (ii) For given $b> -1$, if thewave speed equals $1+b$, then the explicit expressions of thepeakon 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 areobtained. We also verify the correctness of these solutions byusing the software Mathematica. Our work extends some previousresults.