{
  "id": "1307.5284",
  "version": "v2",
  "published": "2013-07-19T17:23:26.000Z",
  "updated": "2015-04-27T13:26:23.000Z",
  "title": "Large time blow up for a perturbation of the cubic Szegő equation",
  "authors": [
    "Haiyan Xu"
  ],
  "comment": "page number:15",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the following Hamiltonian equation on a special manifold of rational functions, $$i\\partial_tu=\\Pi(|u|^2u)+\\alpha (u|1),\\ \\alpha\\in\\mathbb{R},$$ where $\\Pi $ denotes the Szeg\\H{o} projector on the Hardy space of the circle $\\mathbb{S}^1$. The equation with $\\alpha=0$ was first introduced by G\\'erard and Grellier as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\\alpha<0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\\alpha>0$, there exist trajectories on which high Sobolev norms exponentially grow with time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-19T17:23:26.000Z",
      "abstract": "We consider the following Hamiltonian equation on a special manifold of rational functions, \\[i\\p_tu=\\Pi(|u|^2u)+\\al (u|1),\\ \\al\\in\\R,\\] where $\\Pi $ denotes the Szeg\\H{o} projector on the Hardy space of the circle $\\SS^1$. The equation with $\\al=0$ was first introduced by G\\'erard and Grellier in \\cite{GG1} as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\\al<0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\\al>0$, there exist trajectories on which high Sobolev norms exponentially grow with time.",
      "comment": "page number:13",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-27T13:26:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large time blow",
      "high sobolev norms exponentially grow",
      "perturbation",
      "compact subset",
      "totally non dispersive evolution equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5284X"
    }
  }
}