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PreprintarXiv licencearXiv

Renormalization-Group Stability Does Not Determine Front Selection

  • Ko Okumura
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Abstract

The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a paradigmatic example of front propagation and asymptotic-state selection. Using a unified renormalization-group (RG) framework, we show that its traveling-wave solutions form a continuous family of RG fixed points and that all fronts with v >= 2 are locally RG-stable. The resulting RG stability criterion reproduces the classical linear-stability analysis of Canosa, according to which all fronts with v >= 2 are stable. Nevertheless, the asymptotically selected front is v=2, determined by the leading-edge structure of the initial condition rather than by RG stability. The FKPP equation thus provides an explicit example in which RG fixed-point stability does not determine asymptotic-state selection, demonstrating that stability and selection constitute distinct aspects of RG behavior.

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Cite

BibTeXRIS
BibTeX
@misc{okumura2026renormalizationgroup,
  title = {{Renormalization-Group Stability Does Not Determine Front Selection}},
  author = {Ko Okumura},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.02211},
  eprint = {2608.02211},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.02211},
}
RIS
TY  - UNPB
TI  - Renormalization-Group Stability Does Not Determine Front Selection
AU  - Ko Okumura
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2608.02211
UR  - https://arxiv.org/abs/2608.02211
AB  - The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a paradigmatic example of front propagation and asymptotic-state selection. Using a unified renormalization-group (RG) framework, we show that its traveling-wave solutions form a continuous family of RG fixed points and that all fronts with v >= 2 are locally RG-stable. The resulting RG stability criterion reproduces the classical linear-stability analysis of Canosa, according to which all fronts with v >= 2 are stable. Nevertheless, the asymptotically selected front is v=2, determined by the leading-edge structure of the initial condition rather than by RG stability. The FKPP equation thus provides an explicit example in which RG fixed-point stability does not determine asymptotic-state selection, demonstrating that stability and selection constitute distinct aspects of RG behavior.
ER  -