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 there exists a conserved energy which is equivalent to the -norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every .
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
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
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