On the initial value problem for the two-coupled Camassa-Holm system in Besov spaces

Considered herein is the Cauchy problem for the two-coupled Camassa-Holm system. Based on the local well-posedness results for this problem, it is shown that the solution map z(0) -> z(t) of this problem in the periodic case is not uniformly continuous in Besov spaces B(p,)r(s)(T) x B-p,r(s) (T) with s > max{3/2, 1 + 1/ p}, 1 <= p, r <= infinity by using the method of approximate solutions. In the non-periodic case, the nonuniform continuity of this solution map in Besov spaces B-2,r(s) (R) x B-2,r(s) (R) with s > 3/2, 2 <= r <= infinity is established. Finally, the Holder continuity of the solution map in Besov spaces is proved.