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 ∂tv + v∂xv + ∂ 3 x v + δ ( ∂ x v + ∂ x v ) = 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 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 O(1) regime treatable by classical perturbation methods and the & δ regime excluded by parabolic energy estimates. Department of Mathematics, University of Kansas, Lawrence, KS 66045; [email protected]. Research of M.J. was partially supported under NSF grant no. 1211183. Université de Lyon, Université Lyon I, Institut Camille Jordan, UMR CNRS 5208, 43 bd du 11 novembre 1918, F 69622 Villeurbanne Cedex, France; [email protected]: Research of P.N. is partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR CNRS 5208, 43 bd du 11 novembre 1918, F 69622 Villeurbanne Cedex, France; [email protected]: Stay of M.R. in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01 Indiana University, Bloomington, IN 47405; [email protected]: Research of K.Z. was partially supported under NSF grant no. DMS-0300487