{
  "id": "1712.08912",
  "version": "v1",
  "published": "2017-12-24T12:20:18.000Z",
  "updated": "2017-12-24T12:20:18.000Z",
  "title": "Spatial Hamiltonian identities for nonlocally coupled systems",
  "authors": [
    "Berry Bakker",
    "Arnd Scheel"
  ],
  "comment": "38 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-24T12:20:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S30",
      "45G15",
      "35C07",
      "37K05",
      "37L10",
      "37L45"
    ],
    "keywords": [
      "spatial hamiltonian identities",
      "nonlocally coupled systems",
      "euler-lagrange equations",
      "natural hamiltonian formalism",
      "nonlinear integro-differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}