Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics

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.