On the Well-posedness of the Schrödinger-Korteweg-de Vries system

Details On the Well-posedness of the Schrödinger-Korteweg-de Vries system On the Well-posedness of the Schrödinger-Korteweg-de Vries system

2009-11-17

arxiv.org/abs/0911.3197v1

Journal of Differential Equations, Volume 249, Issue 10, 15 November 2010, Pages 2500-2520

Mathematics

Analysis of PDEs

16 pages

Scientific paper

We prove that the Cauchy problem for the Schr\"odinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$ type space, introduced by the first author in \cite{G}, to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares.

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