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 $\partial _t v+v\partial _x v+\partial _x^3 v+\delta \left (\partial _x^2 v +\partial _x^4 v\right )=0,$ $\delta >0$, 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 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 $0<\delta \ll 1$ of the spectrum of the associated linearized operators in a sufficiently small neighborhood of the origin in the spectral plane. This modulational stability analysis is particularly interesting, relying on direct calculations of a reduced periodic Evans function and using in an essential way an analogy with hyperbolic relaxation theory at the level of the associated Whitham modulation equations. A second technical difficulty is the exclusion of high-frequency instabilities lying between the $\mathcal {O}(1)$ regime treatable by classical perturbation methods and the $\gtrsim \delta ^{-1}$ regime excluded by parabolic energy estimates.