Nonuniform Dependence on Initial Data of a Periodic Camassa-Holm System

Abstract

This paper is concerned with some properties of a periodic two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in $H^s$ ($𝕊$), $s>\mathrm{5}/\mathrm{2}$.