∂u/∂t = εu − (1+∇²)²u − u³. A canonical model for pattern formation near a pitchfork bifurcation. Stripes, hexagons, and labyrinthine patterns emerge depending on parameters.
Swift-Hohenberg equation (1977): ∂u/∂t = εu − (q₀²+∇²)²u − u³. For ε>0: band of unstable modes near |k|=q₀. Patterns selected by nonlinearity: stripes (1D), hexagons (subcritical), or labyrinthine. The bifurcation is supercritical → stripe amplitude ∝ √ε. Exact stripe solution: u=A·cos(q₀x) with A=√(4ε/3). Universal normal form for pattern-forming systems (Rayleigh-Bénard, Faraday waves).