M-Shape peakons, dehisced solitons, cuspons and new 1-peak solitons for the Degasperis–Procesi equation

In this paper, we investigate all possible single traveling solitary wave solutions of the Degasperis–Procesi (DP) equation under the boundary condition u → A (A is a constant) as x → ±∞. Regular peakons of the DP equation correspond to the case of A = 0. In the case of A ≠ 0, we find new exact soliton solutions including cuspon, peakon, M-shape peakon, dehisced soliton, and double dehisced 1-peak soliton. In particular, we propose three new types of soliton solutions – M-shape peakon, dehisced soliton, and double dehisced 1-peak soliton, which are given in an explicit form. The most interesting is: for the DP equation the cuspon is a limit of those new peaked solutions solutions. We show some graphs to explain our new solutions.