Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit

Abstract

We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation ${\partial }_{t}v+v{\partial }_{x}v+{\partial }_x^{3}v+\delta ({\partial }_x^{2}v+{\partial }_x^{4}v)=0$, $\delta >0$, in the Korteweg-de Vries limit $\delta \to 0$, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem to the evaluation for each Bloch parameter $\xi \in [0,2\pi ]$ of certain elliptic integrals derived formally (on an incomplete set of frequencies/Bloch parameters, hence as necessary conditions for stability) and numerically evaluated by Bar and Nepomnyashchy \cite{BN}, thus obtaining, up to machine error, complete conclusions about stability. The main technical difficulty is in treating the large-frequency and small Bloch-parameter regimes not studied by Bar and Nepomnyashchy \cite{BN}, which requires techniques rather different from classical Fenichel-type analysis. The passage from small-$\delta$ to small-$\xi$ behavior is particularly interesting, using in an essential way an analogy with hyperbolic relaxation at the level of the Whitham modulation equations.

52 pages, 4 figures

References in corpus (2)

• An index theorem for the stability of periodic traveling waves of KdV type
• Whitham's equations for modulated roll-waves in shallow flows