Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition

We study the time local well-posedness of the Cauchy problem for the modified KdV equation on the one-dimensional torus. We prove that when 1/2 > s > 3/8, it is locally well posed in H
^s
, but the uniformly continuous dependence of solution on initial data breaks down in contrast to the case s ≥ 1/2. This improves the result of Bourgain (1993).