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- W4297083311 abstract "We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ partial_t v+vpartial_x v+partial_x^3 v+delta(partial_x^2 v +partial_x^4 v)=0$, $delta>0$, in the Korteweg-de Vries limit $deltato 0$, 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 $xiin [0,2pi]$ 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." @default.
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- W4297083311 date "2012-02-28" @default.
- W4297083311 modified "2023-09-25" @default.
- W4297083311 title "Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit" @default.
- W4297083311 doi "https://doi.org/10.48550/arxiv.1202.6402" @default.
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