Rigorous pattern selection by invasion for general reaction–diffusion systems with static parameters

Establish a rigorous proof of pattern selection by invasion into an unstable state for general pattern-forming reaction–diffusion systems with a time-independent parameter μ = μ0 > 0 by proving the existence of invasion fronts that select a patterned state in their wake and characterizing the corresponding selection mechanism.

Background

The paper develops formal and numerical predictions for invasion fronts under a slowly increasing temporal parameter in models such as FKPP, complex Ginzburg–Landau, Swift–Hohenberg, and Cahn–Hilliard. For patterned invasion, it predicts front position, leading-edge decay, and the selected wavenumber using a linear analysis and a Burgers-type modulation framework.

Despite these advances, the authors emphasize that a fully rigorous existence and selection theory for pattern selection by invasion remains unresolved even in the simpler static-parameter case (μ constant). They note that proving the existence of pattern-forming invasion fronts and demonstrating wavenumber selection in general systems is still an open problem, with only related partial results currently available.