{
  "id": "1108.2417",
  "version": "v2",
  "published": "2011-08-11T14:31:45.000Z",
  "updated": "2011-10-21T19:15:17.000Z",
  "title": "Linear stability analysis for traveling waves of second order in time PDE's",
  "authors": [
    "Milena Stanislavova",
    "Atanas Stefanov"
  ],
  "comment": "Added some references, proof of Section 3.3 much simplified",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We develop a general theory for linear stability of traveling waves of second order in time PDE's. More precisely, we introduce an explicitly computable index $\\om^*\\in (0, \\infty]$ (depending on the self-adjoint part of the linearized operator) so that the wave is stable if and only if $|c|\\geq \\om^*$. The results are applicable both in the periodic case and in the whole line case. As an application, we consider three classical models - the Boussinesq equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. For the Boussinesq model and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves (and for all powers of $p$ in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander-Sachs, which apply to the case $p=2$. For the beam equation, we provide an explicit formula (depending of the function $\\|\\vp_c'\\|_{L^2}$), which works for all $p$ and for both the periodic and the whole line cases. Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-10-21T19:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear stability analysis",
      "traveling waves",
      "second order",
      "time pdes",
      "kgz system"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/25/9/2625",
      "journal": "Nonlinearity",
      "year": 2012,
      "month": "Sep",
      "volume": 25,
      "number": 9,
      "pages": 2625
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012Nonli..25.2625S"
    }
  }
}