Global fold structure of the Miura map on L2(T)

Abstract

The main purpose of this paper is to study the Miura transform r → r′ + r2-functions. More precisely, we prove that the Miura transform, viewed as map from L2(T) to H−1(T)⁠, has a global fold structure with a “Whitney type” singularity at L02(T)⁠, the space of periodic L2-functions with mean zero. Using the well-known fact that the Miura transform maps solutions of the modified Korteweg-de Vries equation (mKdV) to solutions of the Korteweg-de Vries equation (KdV), the above result can be used as a tool to obtain low-regularity well-posedness results for mKdV on the circle from corresponding low-regularity well-posedness results of KdV (and vice versa).