Kyoto Journal of Mathematics

On the one-dimensional cubic nonlinear Schrödinger equation below $L^{2}$

Tadahiro Oh and Catherine Sulem

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Abstract

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic nonlinear Schrödinger equation (NLS) on the real line $\mathbb {R}$ and on the circle $\mathbb {T}$ for solutions below the $L^{2}$-threshold. We point out common results for NLS on $\mathbb {R}$ and the so-called Wick-ordered NLS (WNLS) on $\mathbb {T}$, suggesting that WNLS may be an appropriate model for the study of solutions below $L^{2}(\mathbb {T})$. In particular, in contrast with a recent result of Molinet, who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from $L^{2}(\mathbb {T})$ to the space of distributions, we show that this is not the case for WNLS.

Article information

Source
Kyoto J. Math. Volume 52, Number 1 (2012), 99-115.

Dates
First available in Project Euclid: 19 February 2012

Permanent link to this document
http://projecteuclid.org/euclid.kjm/1329684744

Digital Object Identifier
doi:10.1215/21562261-1503772

Mathematical Reviews number (MathSciNet)
MR2892769

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Oh, Tadahiro; Sulem, Catherine. On the one-dimensional cubic nonlinear Schrödinger equation below L 2 . Kyoto J. Math. 52 (2012), no. 1, 99--115. doi:10.1215/21562261-1503772. http://projecteuclid.org/euclid.kjm/1329684744.


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