On the Cauchy Problem for the Dispersion Generalized Camassa-Holm Equation

Abstract

In this paper, we establish local well-posedness of the Cauchy problem for a recently proposed dispersion generalized Camassa-Holm equation by using Kato's semigroup approach for quasi-linear evolution equations. We show that for initial data in the Sobolev space $H^s(\mathbb{R})$ with $s>\frac{7}{2}+p$, the Cauchy problem is locally well-posed, where $p$ is an even real number determined by the order of the positive differential operator $L$ corresponding to the dispersive effect added to the Camassa-Holm equation.