Preprint A399/2005
Rough solutions for the periodic Schrödinger - Korteweg-deVries system

Carlos Matheus | Arbieto, Alexander | Corcho, Adán

**Keywords: **
Local and global well-posedness | Schrodinger - Korteweg-deVries system

We prove two new mixed sharp bilinear estimates of Schródinger-Airy type. In
particular, we obtain the local well-posedness of the Cauchy problem of the
Schrodinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}.
Our lowest regularity is $H^{1/4}\times L^2$, which is
somewhat far from the naturally expected endpoint $L^2\times H^{-1/2}$. This is
a novel phenomena related to the periodicity condition. Indeed, in the
continuous case, Corcho and Lineares proved local well-posedness for the
natural endpoint $L^2\times H^{-\frac{3}{4}+}$.
Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the
energy space $H^1\times H^1$ using our local well-posedness result and three
conservation laws discovered by M. Tsutsumi.