  We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 u + u
- u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of
phase slips as a consequence of the {\it local} shape of $u$. We next prove a
{\it global} theorem about evolution from an Eckhaus unstable state, all the
way to the limiting stable finite state, for periodic perturbations of Eckhaus
unstable periodic initial data. Equipped with these results, we proceed to
prove the corresponding phenomena for the fourth order Swift-Hohenberg
equation, of which the Ginzburg-Landau equation is the amplitude approximation.
This sheds light on how one should deal with local and global aspects of phase
slips for this and many other similar systems.
