  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.
