Local Bifurcations from Single-Mode Traveling Waves in the Filamentation Equation
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 Xs. 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.
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BibTeX
@misc{cacchio2026local,
title = {{Local Bifurcations from Single-Mode Traveling Waves in the Filamentation Equation}},
author = {Yuri Cacchiò},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23145},
eprint = {2609.23145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23145},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Bifurcations from Single-Mode Traveling Waves in the Filamentation Equation
AU - Yuri Cacchiò
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23145
UR - https://arxiv.org/abs/2609.23145
AB - 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.
ER -