Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 07, pp. 1-20.

Title: Global well-posedness of NLS-KdV systems for periodic functions

Author: Carlos Matheus (IMPA, Rio de Janeiro, Brazil)

Abstract:
 We prove that the Cauchy problem of the
 Schrodinger-Korteweg-deVries (NLS-KdV) system for
 periodic functions is globally well-posed for initial
 data in the energy space $H^1\times H^1$. More precisely, we
 show that the non-resonant NLS-KdV system is globally well-posed for
 initial data in $H^s(\mathbb{T})\times H^s(\mathbb{T})$
 with $s>11/13$ and the resonant NLS-KdV system is globally well-posed
 with $s>8/9$.
 The strategy is to apply the I-method used by Colliander, Keel,
 Staffilani, Takaoka and Tao. By doing this, we improve the
 results by Arbieto, Corcho and Matheus concerning the global
 well-posedness of  NLS-KdV systems.
 
Submitted November 13, 2006. Published January 2, 2007.
Math Subject Classifications: 35Q55.
Key Words: Global well-posedness; Schrodinger-Korteweg-de Vries system;
           I-method.