On the periodic Cauchy problem for a coupled Camassa–Holm system with peakons

Abstract

In this paper, we first prove that the solution map of the Cauchy problem for a coupled Camassa–Holm system is not uniformly continuous in \({H^{s}(\mathbb{T}) \times H^{s}(\mathbb{T}),s > \frac{3}{2}}\), the proof of which is based on well posedness estimates and the method of approximate solutions. Then we study the continuity properties of its solution map further and show that it is Holder continuous in the \({H^\sigma(\mathbb{T}) \times H^\sigma(\mathbb{T})}\) topology with \({\frac{1}{2} < \sigma < s}\). Our results can also be carried out on the nonperiodic case.