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

Abstract

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 1 < s <= 3/2 is established under the assumptions u0 ε Hs and t norm of ||u0x||L∞ < ∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with s > 3/2 is also developed.