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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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