Subharmonic Dynamics of Wave Trains in the Korteweg-de Vries / Kuramoto-Sivashinsky Equation

Abstract

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).