Front Selection Is Not Determined by Renormalization-Group Stability

The Fisher--Kolmogorov--Petrovsky--Piskunov 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 $v \ge 2$ are RG-stable. Nevertheless, the asymptotically selected front is not determined by RG stability. The FKPP equation therefore provides an explicit example in which RG fixed-point stability and asymptotic-state selection are distinct concepts.