{
  "id": "1202.6402",
  "version": "v1",
  "published": "2012-02-28T22:37:25.000Z",
  "updated": "2012-02-28T22:37:25.000Z",
  "title": "Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit",
  "authors": [
    "Mathew A. Johnson",
    "Pascal Noble",
    "L. Miguel Rodrigues",
    "Kevin Zumbrun"
  ],
  "comment": "52 pages, 4 figures",
  "categories": [
    "math.AP",
    "math.CA",
    "math.SP"
  ],
  "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(\\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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-28T22:37:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "korteweg-de vries/kuramoto-sivashinsky equation",
      "korteweg-de vries limit",
      "periodic wave trains",
      "spectral stability",
      "unstable thin film flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.6402J"
    }
  }
}