Subharmonic dynamics of wave trains in reaction–diffusion systems

Highlights: • We present nonlinear stability of periodic solutions to reaction diffusion equations. • We consider subharmonic perturbations. • Stability result is uniform in period of perturbation. • Methodology presented here applies to more general dissipative systems We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction–diffusion systems. For each $N\in N$, 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.