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


Kappeler, T; Topalov, P (2004). Global fold structure of the Miura map on L2(T). International Mathematics Research Notices, (39):2039-2068.

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 L20(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).

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 L20(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).

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Additional indexing

Other titles:
Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2004
Deposited On:18 Feb 2010 14:01
Last Modified:05 Apr 2016 13:24
Publisher:Oxford University Press
ISSN:1073-7928
Publisher DOI:https://doi.org/10.1155/S1073792804133205
Official URL:http://imrn.oxfordjournals.org
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2062735

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