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 H 1 -norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space H s with 1 < s ⩽ 3 2 is established under the assumptions u 0 ∈ H s and ‖ u 0 x ‖ L ∞ < ∞ . The local well-posedness of solutions for the equation in the Sobolev space H s ( R ) with s > 3 2 is also developed.