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 $1<s⩽\frac{3}{2}$ is established under the assumptions $u_0\in H^s$ and $\Vert u_{0x}{\Vert }_{L^{\infty }}<\infty$. The local well-posedness of solutions for the equation in the Sobolev space $H^s(R)$ with $s>\frac{3}{2}$ is also developed.