Linear and nonlinear front instabilities in bistable systems

Abstract

The stability of planar fronts to transverse perturbations in bistable systems is studied using the Swift–Hohenberg model and an urban population model. Contiguous to the linear transverse instability that has been studied in earlier works, a parameter range is found where planar fronts are linearly stable but nonlinearly unstable; transverse perturbations beyond some critical size grow rather than decay. The nonlinear front instability is a result of the coexistence of stable planar fronts and stable large-amplitude patterns. While the linear transverse instability leads to labyrinthine patterns through fingering and tip splitting, the nonlinear instability often evolves to spatial mixtures of stripe patterns and irregular regions of the uniform states.