Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit
Résumé
We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation v_t + v v_x + v_xxx + δ(v_xx+v_xxxx) = 0, δ > 0, in the Korteweg-de Vries limit δ → 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 of spectral stability in this limit to the validation of a set of three conditions, each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non-empty set of parameter space. The main technical difficulty in our analysis, and one that has not been previously addressed by any authors, is that of obtaining a useful description for 0 < δ<>1/δ regime excluded by parabolic energy estimates.
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