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The Miura map on the line


Kappeler, T; Perry, P; Shubin, M; Topalov, P (2005). The Miura map on the line. International Mathematics Research Notices, 2005(50):3091-3133.

Abstract

Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Korteweg-de Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura

Abstract

Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Korteweg-de Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Language:English
Date:2005
Deposited On:18 Feb 2010 14:07
Last Modified:23 Jan 2022 14:35
Publisher:Oxford University Press
ISSN:1073-7928
Additional Information:This is a pre-copy-editing, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version "Kappeler, T; Perry, P; Shubin, M; Topalov, P (2005). The Miura map on the line. International Mathematics Research Notices, 2005(50):3091-3133" is available online at:10.1155/IMRN.2005.3091
OA Status:Green
Publisher DOI:https://doi.org/10.1155/IMRN.2005.3091
Related URLs:http://arxiv.org/abs/math/0506411
  • Content: Accepted Version