Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H s ( T ) × H r ( T ) to C ( [ 0 , 1 ] , H s ( T ) × H r ( T ) ) when s ⩾ 1 and r ⩾ 0 . We also demonstrate the nonuniform continuous property in the continuous function space C 1 ( T ) × C 1 ( T ) .