Global N-peakon weak solutions to a family of nonlinear equations

We use a double mollification method to study a family of high order nonlinear partial differential equations with peakon weak solutions. This method gives the approximated N-peakon weak solutions which satisfy weak consistent property. By a limiting process, we prove the global existence of N-peakon weak solutions. Moreover, the traveling speeds of single peakon weak solutions (soliton waves) are obtained directly by the limit of the mollification.