{
  "id": "1204.0756",
  "version": "v3",
  "published": "2012-04-03T18:19:17.000Z",
  "updated": "2013-03-18T16:09:43.000Z",
  "title": "Integrability of higher pentagram maps",
  "authors": [
    "Boris Khesin",
    "Fedor Soloviev"
  ],
  "comment": "46 pages, 4 figures; scaling for even dimensions is corrected in section 8",
  "categories": [
    "math.DS",
    "math.SG",
    "nlin.SI"
  ],
  "abstract": "We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartz's definition of the 2D pentagram map. We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We also study in detail the 3D case, where we prove integrability for both closed and twisted polygons and describe the spectral curve, first integrals, the corresponding tori and the motion along them, as well as an invariant symplectic structure.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-03-18T16:09:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integrability",
      "define higher pentagram maps",
      "invariant symplectic structure",
      "scale invariant maps",
      "2d pentagram map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.0756K"
    }
  }
}