We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction–diffusion systems. For each $N\in \mathbb{N}$, such $T$-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to $NT$-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on $N$, and, in particular, they tend to zero as $N\to \infty$, leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in $N$. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.

我们研究了反应扩散系统中谱稳定波列的稳定性和非线性局部动力学。对于每个$ñ\in \mathbb{ñ}$， 这样的 $Ť$周期行波很容易被看作是非线性渐近稳定的（具有渐近相位），当受到 $ñŤ$-周期（即次谐波）扰动。但是，扰动的允许大小和衰减的指数速率都取决于$ñ$，尤其是当它们趋于零时， $ñ\to \infty$，导致这种次谐波稳定性结果缺乏一致性。在这项工作中，我们以作者最近的工作为基础，并介绍了一种方法，可以使我们获得亚谐波扰动的稳定结果，该结果在$ñ$。当受到局部扰动（即在线上可积分）的扰动时，这种波的动力激发了我们的工作，最近，这种扰动引起了许多作者的关注。