Korteweg-de Vries / Kuramoto-Sivashinsky方程中波列的亚谐动力学
Subharmonic Dynamics of Wave Trains in the Korteweg-de Vries / Kuramoto-Sivashinsky Equation
摘要：
研究了Korteweg-de Vries / Kuramoto-Sivashinsky方程在周期扰动下谱稳定周期波列的稳定性和非线性局部动力学.已知对于每个$N\in\mathbb{N}$，这样的$T$ -周期波训练对$NT$ -周期渐近稳定，即，次谐波，扰动，在这个意义上说，最初附近的数据将渐近收敛到一个小的伽利略推动的基础波，与指数衰减率。然而，允许的初始扰动的大小和衰减的指数速率取决于$N$，事实上，往往为零的$N\to\infty$，导致缺乏均匀性，在这样的分谐波稳定性的结果。我们在这里的目标是建立在最近的方法介绍了作者在反应扩散设置，并实现一个分谐波稳定性的结果是均匀的$N$。这项工作的动机是这样的波列的动力学时，受到扰动是局部的（即，线上可积）。
英文摘要：
We study the stability and nonlinear local dynamics of spectrally stable periodic wave trains of the Korteweg-de Vries / Kuramoto-Sivashinsky equation when subjected to classes of periodic perturbations. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is asymptotically stable to $NT$-periodic, i.e., subharmonic, perturbations, in the sense that initially nearby data will converge asymptotically to a small Galilean boost of the underlying wave, with exponential rates of decay. However, both the allowable size of initial perturbations and the exponential rates of decay depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in such subharmonic stability results. Our goal here is to build upon a recent methodology introduced by the authors in the reaction-diffusion setting and achieve a subharmonic stability result which is uniform in $N$. This work is motivated by the dynamics of such wave trains when subjected to perturbations which are localized (i.e., integrable on the line).
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