Exact peakon solutions given by the generalized hyperbolic functions for some nonlinear wave equations

In 1993, Camassa and Holm drived a shallow water equation and found that this equation has a peakon solution with the form $phi(xi)=ce^{-|xi|}$. In this paper, we show that three nonlinear wave systems have peakon solutions which needs to be represented as generalized hyperbolic functions. For the existence of these solutions, some constraint parameter conditions are derived.