Renormalization-Group Stability Does Not Determine Front Selection

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.

arXiv comment8 pages, no figure