03-43 Guillaume van Baalen
Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics. (364K, LaTeX2e with 3 ps figures) Feb 10, 03
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Abstract. 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 $s\approx-\frac{1}{2}~\eta''$ 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 $$\partial_t\eta= -\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)~\triangle^2\eta -\epsilon^2\triangle\eta -{(1+\alpha^2)}~(\eta')^2$$ for times $t_0\leq{\cal O}(\epsilon^{-52/5}~L_0^{-32/5})$, while the amplitude $1+\alpha^2~s$ is essentially constant.

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