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

TL;DR

This paper investigates the stability of periodic wave trains in the Korteweg-de Vries / Kuramoto-Sivashinsky equation, aiming to establish uniform subharmonic stability results that are independent of the perturbation period parameter.

Contribution

The authors extend recent reaction-diffusion methods to prove uniform subharmonic stability of wave trains in the KdV/Kuramoto-Sivashinsky equation, regardless of the perturbation period.

Findings

Established uniform stability results for wave trains under subharmonic perturbations.

Demonstrated decay rates depend on the perturbation period but can be made uniform.

Provided insights into localized perturbations and their effects on wave train stability.

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).