{
  "id": "1408.0494",
  "version": "v1",
  "published": "2014-08-03T12:53:20.000Z",
  "updated": "2014-08-03T12:53:20.000Z",
  "title": "A note on the existence of traveling-wave solutions to a Boussinesq system",
  "authors": [
    "Filipe Oliveira"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We obtain a one-parameter family $$(u_{\\mu}(x,t),\\eta_{\\mu}(x,t))_{\\mu\\geq \\mu_0}=(\\phi_{\\mu}(x-\\omega_{\\mu} t),\\psi_{\\mu}(x-\\omega_{\\mu} t))_{\\mu\\geq \\mu_0}$$ of traveling-wave solutions to the Boussinesq system $$u_t+\\eta_x+uu_x+c\\eta_{xxx}=0,\\eta_t+u_x+(\\eta u)_x+au_{xxx}=0$$ in the case $a,c<0$, with non-null speeds $\\omega_{\\mu}$ arbitrarily close to $0$ ($\\omega_{\\mu}\\xrightarrow[\\mu\\to+\\infty]{} 0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $\\mu$) estimate $\\|\\phi_{\\mu}\\|_2^2+\\|\\psi_{\\mu}\\|_2^2\\leq C\\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, $\\phi_{\\mu}$ and $-\\psi_{\\mu}$ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-03T12:53:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "traveling-wave solutions",
      "boussinesq system",
      "non-null speeds",
      "traveling-waves satisfies",
      "one-parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}