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T. Kappeler, P. Topalov, Global fold structure of the Miura map on , International Mathematics Research Notices, Volume 2004, Issue 39, 2004, Pages 2039–2068, https://doi.org/10.1155/S1073792804133205
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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 to , has a global fold structure with a “Whitney type” singularity at , 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).