  We consider the Hirota-Satsuma system, a coupled KdV-type system, with
periodic boundary conditions. The first part of the paper concerns with the
smoothing estimates for the system. More precisely, it is shown that, for
initial data in a Sobolev space, the difference of the nonlinear and linear
evolutions lies in a smoother space. The smoothing gain we obtain depends very
much on the arithmetic nature of the coupling parameter $a$ which determines
the structure of the resonant sets in the estimates. In the second part, we
address the forced and damped Hirota-Satsuma system and obtain counterpart
smoothing estimates. As a consequence of these estimates, we prove the
existence and smoothness of a global attractor in the energy space.
