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

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 $H^1imes H^1$. More precisely, we show that the non-resonant NLS-KdV system is globally well-posed for initial data in $H^s(mathbb{T})imes H^s(mathbb{T})$ 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