A model containing both the Camassa–Holm and Degasperis–Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa–Holm and Degasperis–Procesi equations as special cases, is investigated. Although the H1-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space Hs with is established under the assumptions u0∈Hs and ‖u0x‖L∞<∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with is also developed.