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Global fold structure of the Miura map on $$L^2({\mathbb{T}})$$. (English) Zbl 1076.35111
Summary: The main purpose of this paper is to study the Miura transform $$r\to r'+ r^2$$ on periodic $$L^2$$-functions. More precisely, we prove that the Miura transform, viewed as map from $$L^2(\mathbb{T})$$ to $$H^{-1}(\mathbb{T})$$, has a global fold structure with a ‘Whitney type’ singularity at $$L^2(\mathbb{T})$$, the space of periodic $$L^2$$-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).

MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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