Spectral stability of periodic wave trains of the Korteweg-de   Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit
  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. 
Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin   Zumbrun
math.AP math.CA math.SP