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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.