{
  "abstract": "We study local bifurcations from the single-mode traveling wave family of the filamentation equation on the torus, within the real positive-frequency Sobolev space $X^s$. A Fourier null structure removes the apparent derivative loss and makes the traveling wave map real analytic for $s>3/2$. Its linearization splits into finitely many coupled two-mode blocks, a carrier block, and a diagonal high-frequency tail. For negative temporal frequency, we determine the critical values generated by both the finite coupled blocks and the tail. At every non resonant finite-block critical value, Lyapunov-Schmidt reduction yields a locally unique real-analytic branch, with vanishing linear correction to the bifurcation parameter. Moreover, each finite tail critical value produces a simple local branch. These values accumulate at a parameter where the linearization ceases to be Fredholm. The accumulation point is nevertheless a bifurcation point in the usual topological sense. For $σ=1$, a symmetry in the first Fourier mode produces an exact two-mode vertical branch, including the exceptional non-Fredholm case $k=2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23145",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuri Cacchiò"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/31tmpfb8csz0n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23145",
  "fulltext_url": null,
  "id": "31tmpfb8csz0n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23145",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7214034933",
  "pdf_url": "https://arxiv.org/pdf/2609.23145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/31tmpfb8csz0n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local Bifurcations from Single-Mode Traveling Waves in the Filamentation Equation",
  "title_text": "Local Bifurcations from Single-Mode Traveling Waves in the Filamentation Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/31tmpfb8csz0n",
  "venue": "arXiv",
  "versions": []
}