Bistable reaction-diffusion equation on a 2D periodic domain:
∂u/∂t − D·Δu = α·u·(1−u²).
Stationary points: u = 0 (unstable), u = ±1
(stable). Random initial perturbation grows into bistable
domains separated by sharp fronts; the diffusion smooths the
fronts, the reaction sharpens them.
Slow / fast separation: the reaction term dominates inside each
domain (locks u to ±1 fast); the diffusion term dominates at the
fronts (slow front motion governed by the curvature of the
domain boundary). The phase-separation pattern that emerges is
the Allen-Cahn / Cahn-Hilliard kinetics that underlies many
physical pattern-formation processes (alloy spinodal
decomposition, magnetic domain coarsening, etc.).
Red: u > 0 domains. Blue: u < 0 domains.
Black polyline: cross-cut of u along j = N/2.