Guillaume van Baalen
Phase turbulence in the Complex Ginzburg--Landau equation 
via Kuramoto--Sivashinsky phase dynamics.
(364K, LaTeX2e with 3 ps figures)

ABSTRACT.  We study the Complex Ginzburg--Landau initial 
value problem 
\begin{equation} 
\partial_t u=(1+i\alpha)~\partial_x^2 u + u - (1+i\beta)~u~|u|^2~, 
u(x,0)=u_0(x) 
\end{equation} 
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 $s\approx-\frac{1}{2}~\eta''$ 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 
\begin{equation} 
\partial_t\eta= 
-\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)~\triangle^2\eta 
-\epsilon^2\triangle\eta 
-{(1+\alpha^2)}~(\eta')^2 
\end{equation} 
for times $t_0\leq{\cal O}(\epsilon^{-52/5}~L_0^{-32/5})$, 
while the amplitude $1+\alpha^2~s$ is essentially constant.