{
  "id": "1804.11155",
  "version": "v1",
  "published": "2018-04-30T12:31:06.000Z",
  "updated": "2018-04-30T12:31:06.000Z",
  "title": "Unique Determination of Sound Speeds for Coupled Systems of Semi-linear Wave Equations",
  "authors": [
    "Alden Waters"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider coupled systems of semi-linear wave equations with different sound speeds on a finite time interval $[0,T]$ and a bounded Lipschitz domain $\\Omega$ in $\\mathbb{R}^3$, with boundary $\\partial\\Omega$. We show the coupled systems are well posed for variable coefficient sounds speeds and short times. Under the assumption of small initial data, we prove the Dirichlet-to-Neumann map on $[0,T]\\times\\partial\\Omega$ associated with the nonlinear problem is sufficient to determine the Dirichlet-to-Neumann map for the linear problem. We can then reconstruct the sound speeds in $\\Omega$ for the coupled nonlinear wave equations under certain geometric assumptions. In the case of the full source to solution map in $\\Omega\\times[0,T]$ this reconstruction could also be accomplished under fewer geometric assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-30T12:31:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semi-linear wave equations",
      "sound speeds",
      "coupled systems",
      "unique determination",
      "dirichlet-to-neumann map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}