{
  "id": "1605.03472",
  "version": "v1",
  "published": "2016-05-11T15:17:40.000Z",
  "updated": "2016-05-11T15:17:40.000Z",
  "title": "A sufficient condition for a Rational Differential Operator to generate an Integrable System",
  "authors": [
    "Sylvain Carpentier"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.RA"
  ],
  "abstract": "For a rational differential operator $L=AB^{-1}$, the Lenard-Magri scheme of integrability is a sequence of functions $F_n, n\\geq 0$, such that (1) $B(F_{n+1})=A(F_n)$ for all $n \\geq 0$ and (2) the functions $B(F_n)$ pairwise commute. We show that, assuming that property $(1)$ holds and that the set of differential orders of $B(F_n)$ is unbounded, property $(2)$ holds if and only if $L$ belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator $L$ is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence $(F_n)$ starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-11T15:17:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational differential operator",
      "sufficient condition",
      "integrable system",
      "rational operator",
      "differential orders"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}