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.