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15 November 2018 Conserved energies for the cubic nonlinear Schrödinger equation in one dimension
Herbert Koch, Daniel Tataru
Duke Math. J. 167(17): 3207-3313 (15 November 2018). DOI: 10.1215/00127094-2018-0033

Abstract

We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each s>12 there exists a conserved energy which is equivalent to the Hs-norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every s1.

Version Information

The current version of this article supersedes the original advance publication version posted on 26 October 2018. Corrections have been made in the following locations: equations (1.2), (2.2), (2.4), and (2.11); the displays in the proof of Lemma 4.3; the last two displays in the proof of Proposition B.2; the second display in the proof of Theorem B.18; and the third paragraph in Appendix C.

Citation

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Herbert Koch. Daniel Tataru. "Conserved energies for the cubic nonlinear Schrödinger equation in one dimension." Duke Math. J. 167 (17) 3207 - 3313, 15 November 2018. https://doi.org/10.1215/00127094-2018-0033

Information

Received: 12 August 2017; Revised: 23 May 2018; Published: 15 November 2018
First available in Project Euclid: 26 October 2018

zbMATH: 07000594
MathSciNet: MR3874652
Digital Object Identifier: 10.1215/00127094-2018-0033

Subjects:
Primary: 35Q55
Secondary: 35Q53 , 37K10

Keywords: fractional Sobolev bounds , Korteweg-de Vries , new conserved energies , nonlinear Schrödinger , transmission coefficient

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 17 • 15 November 2018
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