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

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 $u\in \mathbf{C}$, with $\alpha ,\beta \in \mathbf{R}$. We consider the Benjamin--Feir linear instability region $1+\alpha \beta =-{\epsilon }^2$ with $\epsilon \ll 1$ and ${\alpha }^2<1/2$. We show that for all $\epsilon \le \mathcal{O}(\sqrt{1-2{\alpha }^2}{L}_0^{-32/37})$, and for all initial data $u_0$ sufficiently close to 1 (up to a global phase factor $\ed^{i ϕ_0}, ϕ_0\in{\bf R}$) 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 $s,\eta$ have bounded norms in various ${Ł}^p$ and Sobolev spaces. We prove that $s\approx -1/2{\eta }^{\prime \prime }$ apart from $\mathcal{O}({\epsilon }^2)$ corrections whenever the initial data satisfy this condition, and that in the linear instability range $L_0^{-1}\le \epsilon \le \mathcal{O}(L_0^{-32/37})$, 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 ${\partial }_t\eta =-\left(\frac{1+{\alpha }^2}{2}\right){△}^2\eta -{\epsilon }^2△\eta -(1+{\alpha }^2)({\eta }^{\prime }{)}^2$ for times $t_0\le \mathcal{O}({\epsilon }^{-52/5}{L}_0^{-32/5})$, while the amplitude $1+{\alpha }^{2}s$ is essentially constant.

References in corpus (1)

• Wound-up phase turbulence in the Complex Ginzburg-Landau equation