A new integrable equation with peakons and cuspons and its bi-Hamiltonian structure

A new hierarchy of integrable nonlinear evolution equations which contains the generalized Harry Dym type equation is proposed. The Lax pair representation and bi-Hamilton structures of this hierarchy are derived. Based on travelling wave solution approach, both implicit and explicit smooth solitons, peakon, cuspon, periodic solutions and (anti-)kink solutions of the generalized Harry Dym type equation are obtained.