复Ginzburg-Landau方程中的相位turbulence与Kuramoto-Sivashinsky相位动力学
Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics
摘要：
本文研究复Ginzburg-Landau初值问题$\partial_t u=(1+iα) \partial_x^2 u + u - (1+iβ) u |u|^2$，$u(x,0)=u_0(x)$对复域$u\in{\bf C}$，用$α,β\in{\bf R}$ .本文用$ε\ll1$和$α^2<1/2$研究了Benjamin-Feir线性不稳定区$1+αβ=-ε^2$。我们表明，对于所有的$ε\leq{\cal O}(\sqrt{1-2α^2} L_0^{-32/37})$，并为所有的初始数据$u_0$充分接近1（高达全球相位因子$\ed^{i ϕ_0}, ϕ_0\in{\bf R}$）在适当的空间，存在一个独特的（空间）空间周期解$L_0$。这些解是行波解的{ \em偶}小扰动，$u=(1+α^2 s) \ed^{i ϕ_0-iβt} \ed^{iαη}$和$s,η$在各种$Ł^p$和Sobolev空间中具有有界范数。我们证明，$s\approx-{1/2} η''$除了${\cal O}(ε^2)$校正时，初始数据满足这个条件，并在线性不稳定范围$L_0^{-1}\leqε\leq{\cal O}(L_0^{-32/37})$，动力学基本上是由运动的阶段单独，因此表现出“相序”。事实上，我们证明，相位$η$满足Kuramoto-Sivashinsky方程$\partial_tη= -\bigl({\textstyle\frac{1+α^2}{2}}\bigr) \triangle^2η-ε^2\triangleη-{(1+α^2)} (η')^2$的时间$t_0\leq{\cal O}(ε^{-52/5} L_0^{-32/5})$，而幅度$1+α^2 s$基本上是恒定的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
