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In this paper we study weak continuity of the dynamical systems for the KdVequation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topicshould have significant applications in the study of other properties of theseequations such as finite time blow-up and asymptotic stability and instabilityof solitary waves. The spaces considered here are borderline Sobolev spaces forthe corresponding equations from the viewpoint of the local well-posednesstheory. We first use a variant of the method of [5] to prove weak continuityfor the mKdV, and next use a similar result for a mKdV system and thegeneralized Miura transform to get weak continuity for the KdV equation.

Weak Continuity of Dynamical Systems for the KdV and mKdV Equations