∂ₜu = [ε − (1+∇²)²]u − u³ — stripes, hexagons & squares from noise
The Swift-Hohenberg equation is the normal form for pattern-forming instabilities. For ε<0 the flat state is stable; at ε=0 modes at wavenumber |k|=1 go unstable. The cubic term −u³ saturates growth. Depending on initial conditions and ε, the system selects stripes (1D rolls), hexagons, or disordered labyrinthine patterns — a minimal model for Rayleigh-Bénard convection, reaction-diffusion, and optical patterns.