Well-posedness for the Cauchy problem associated to the Hirota-Satsuma equation: Periodic case

We consider a system of Korteweg-de Vries (KdV) equations coupled through nonlinear terms, called the Hirota-Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restriction norm method, we prove local well-posedness whenever s>−1/2. Also, with some restriction on the parameters of the system, we use the recent technique introduced by Colliander et al., called I-method and almost conserved quantities, to prove global well-posedness for s>−3/14.