English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We study the spectral stability of a family of periodic wave trains of theKorteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_xv+\partial_x^3 v+\delta(\partial_x^2 v +\partial_x^4 v)=0$, $\delta>0$, in theKorteweg-de Vries limit $\delta\to 0$, a canonical limit describingsmall-amplitude weakly unstable thin film flow. More precisely, we carry out arigorous singular perturbation analysis reducing the problem to the evaluationfor each Bloch parameter $\xi\in [0,2\pi]$ of certain elliptic integralsderived formally (on an incomplete set of frequencies/Bloch parameters, henceas necessary conditions for stability) and numerically evaluated by Bar andNepomnyashchy \cite{BN}, thus obtaining, up to machine error, completeconclusions about stability. The main technical difficulty is in treating thelarge-frequency and small Bloch-parameter regimes not studied by Bar andNepomnyashchy \cite{BN}, which requires techniques rather different fromclassical Fenichel-type analysis. The passage from small-$\delta$ tosmall-$\xi$ behavior is particularly interesting, using in an essential way ananalogy with hyperbolic relaxation at the level of the Whitham modulationequations.

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