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

We study the Complex Ginzburg--Landau initial value problem $\partial_t u=(1+iα) \partial_x^2 u + u - (1+iβ) u |u|^2$, $u(x,0)=u_0(x)$ for a complex field $u\in{\bf C}$, with $α,β\in{\bf R}$. We consider the Benjamin--Feir linear instability region $1+αβ=-ε^2$ with $ε\ll1$ and $α^2<1/2$. We show that for all $ε\leq{\cal O}(\sqrt{1-2α^2} L_0^{-32/37})$, and for all initial data $u_0$ sufficiently close to 1 (up to a global phase factor $\ed^{i ϕ_0}, ϕ_0\in{\bf R}$) 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, $u=(1+α^2 s) \ed^{i ϕ_0-iβt} \ed^{iαη}$, and $s,η$ have bounded norms in various $Ł^p$ and Sobolev spaces. We prove that $s\approx-{1/2} η''$ apart from ${\cal O}(ε^2)$ corrections whenever the initial data satisfy this condition, and that in the linear instability range $L_0^{-1}\leqε\leq{\cal O}(L_0^{-32/37})$, the dynamics is essentially determined by the motion of the phase alone, and so exhibits `phase turbulence'. Indeed, we prove that the phase $η$ satisfies the Kuramoto--Sivashinsky equation $\partial_tη= -\bigl({\textstyle\frac{1+α^2}{2}}\bigr) \triangle^2η-ε^2\triangleη-{(1+α^2)} (η')^2$ for times $t_0\leq{\cal O}(ε^{-52/5} L_0^{-32/5})$, while the amplitude $1+α^2 s$ is essentially constant.