SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5389-5421, January 2020.
We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg--Landau equation $\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi |\Psi|^2$ near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters $ \alpha $, $ \beta $, a number of modulation equations can be derived, such as the KdV equation, the Cahn--Hilliard equation, and a family of Ginzburg--Landau based amplitude equations. Here we establish error estimates showing that the Korteweg--de Vries (KdV) approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping, we work in spaces of analytic functions.

中文翻译：

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Eckhaus边界附近的调制方程：KdV方程

SIAM数学分析杂志，第52卷，第6期，第5389-5421页，2020年1月。
我们对复杂的Ginzburg-Landau方程$ \ partial_T \ Psi =（1+ i \ alpha）\ partial_X ^ 2 \ Psi + \ Psi-（（ 1 + i \ beta）\ Psi | \ Psi | ^ 2 $靠近Eckhaus边界，也就是说，当波列接近其第一次不稳定的阈值时。根据参数$ \ alpha $，$ \ beta $，可以得出许多调制方程，例如KdV方程，Cahn-Hilliard方程以及基于Ginzburg-Landau的振幅方程。在这里，我们建立了误差估计，表明Korteweg-de Vries（KdV）近似在特定参数范围内可以做出正确的预测。我们的证明基于能量估计，并利用了临界模式的守恒律结构。为了改善线性阻尼，