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

TL;DR

This paper proves local well-posedness for a generalized Camassa-Holm equation with dispersion, using Kato's semigroup approach, for initial data in certain Sobolev spaces.

Contribution

It establishes the local well-posedness of a dispersion generalized Camassa-Holm equation using Kato's method, extending previous results to include dispersive effects.

Findings

Well-posedness for initial data in H^s with s > 7/2 + p

Application of Kato's semigroup approach to a new class of equations

Extension of Camassa-Holm theory to dispersive generalizations

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.