On peaked and smooth solitons for the Camassa-Holm equation

This letter presents all possible explicit single soliton solutions for the Camassa-Holm (CH) equation mt + mxu + 2mux = 0, m = u − uxx. This equation is studied under the boundary condition u → A (A is a constant) as x → ±∞. Regular peakon solutions correspond to the case of A = 0. For the case of A ≠ 0, both new peaked solitons and new type of smooth solitons, which are expressed in terms of trigonometric and hyperbolic functions, are tremendously given through investigating a Newton equation with a new potential. Mathematical analysis and numeric graphs are provided for those smooth soliton and new peaked soliton solutions.