{
  "id": "2410.01494",
  "version": "v2",
  "published": "2024-10-02T12:44:50.000Z",
  "updated": "2025-01-02T18:52:50.000Z",
  "title": "Equilibrium of Charges and Differential Equations Solved by Polynomials II",
  "authors": [
    "Igor Loutsenko",
    "Oksana Yermolayeva"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in [20]. Although for two species of vortices with circulation ratio -1 the relationship between the equilibria and the factorization/Darboux transformation of the Schrodinger operator was established a long ago, the question about similar relationship for the ratio -2 remained unanswered. Here we present the answer: One has to consider Darboux-type transformations of third order differential operators rather than second order Schrodinger operators. Furthermore, we show that such transformations can also generate equilibrium configurations where an additional charge of a third specie is present. Relationship with integrable hierarchies is briefly discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-01-02T18:52:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differential equations",
      "polynomials",
      "third order differential operators",
      "second order schrodinger operators",
      "2d ideal fluid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}