{
  "annotation_id": "7f9865a0-7e99-4931-b102-505615e80c9e",
  "date_created": "2026-03-02T18:00:28.846000Z",
  "date_modified": "2026-03-02T18:00:28.846000Z",
  "file_hash": "179e6356858a76d3dfbfaddf81451e867ace33bb32ec982ebb695e061c120200",
  "private": false,
  "record": {
    "abstract": "Assuming that a formal approximation of multiple waves has been obtained by\nmatched asymptotic methods, we derive a {\\em Spatial Shadowing lemma} to\nconstruct exact solutions near the formal approximation.\n  In Part I, we consider a general singularly perturbed parabolic system. $$\n\\epsilon u_t + (-\\epsilon^2)^m D^{2m}_x u = f(u,\\epsilon u_x,\\cdots,(\\epsilon\nD_x)^{2m-1} u,x,\\epsilon). $$ We show that if the formal approximation is\nprecise, there is always an exact solution nearby for at least a short time.\nExamples include Cahn-Hilliard equation and viscous profile of conservation\nlaws.\n  In Part II, we show under some more assumptions, the process in Part I can be\nrepeated to obtain global solutions if the formal approximation is a global\none. Examples include reaction-diffusion equations and phase field equations.",
    "arxiv_id": "patt-sol/9601003",
    "authors": [
      "Xiao-Biao Lin"
    ],
    "categories": [
      "patt-sol",
      "nlin.PS"
    ],
    "journal_ref": "Progress in Nonlinear Diff. Equations Their Appl. 19 (1996)\n  385-404",
    "title": "Local and Global Existence of Multiple Waves Near Formal Approximations",
    "url": "https://arxiv.org/abs/patt-sol/9601003"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "d5dcafe6-38aa-42a8-a8d7-4b80ff5fd75c",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}