  We study the Complex Ginzburg--Landau initial value problem $\partial_t
u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a
complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the
Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with
$\epsilon\ll1$ and $\alpha^2<1/2$.
  We show that for all $\epsilon\leq{\cal O}(\sqrt{1-2\alpha^2} L_0^{-32/37})$,
and for all initial data $u_0$ sufficiently close to 1 (up to a global phase
factor $\ed^{i \phi_0}, \phi_0\in{\bf R}$) in the appropriate space, there
exists a unique (spatially) periodic solution of space period $L_0$.
  These solutions are small {\em even} perturbations of the traveling wave
solution, $u=(1+\alpha^2 s) \ed^{i \phi_0-i\beta t} \ed^{i\alpha \eta}$, and
$s,\eta$ have bounded norms in various $\L^p$ and Sobolev spaces.
  We prove that $s\approx-{1/2} \eta''$ apart from ${\cal O}(\epsilon^2)$
corrections whenever the initial data satisfy this condition, and that in the
linear instability range $L_0^{-1}\leq\epsilon\leq{\cal O}(L_0^{-32/37})$, the
dynamics is essentially determined by the motion of the phase alone, and so
exhibits `phase turbulence'.
  Indeed, we prove that the phase $\eta$ satisfies the Kuramoto--Sivashinsky
equation $\partial_t\eta= -\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)
\triangle^2\eta -\epsilon^2\triangle\eta -{(1+\alpha^2)} (\eta')^2$ for times
$t_0\leq{\cal O}(\epsilon^{-52/5} L_0^{-32/5})$, while the amplitude
$1+\alpha^2 s$ is essentially constant.
