  We consider the periodic non-linear Schr\"odinger equation with non-linearity
given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that
the difference between the non-linear evolution and a phase rotation of the the
linear evolution is in a smoother space. We then study forced and damped
defocusing non-linear Schr\"odinger equations of the above type and establish
an analogous smoothing statement that extends globally in time. As a corollary
we establish both existence and smootheness for global attractors in the energy
space.
