Non-uniform dependence on initial data for the CH equation on the line

Abstract

For $s > 1$ two sequences of CH solutions living in a bounded subset of the Sobolev space $H^s(\mathbb{R})$ are constructed, whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant. This implies that the solution map of the CH equation is not uniformly continuous in $H^s(\mathbb{R})$.