∂ₜu = εu − (1+∇²)²u − u³ — stripes, hexagons, spirals from a PDE
Swift-Hohenberg (1977) was derived to model convective instabilities. The operator (1+∇²)² penalizes deviations from critical wavenumber k=1, selecting a natural wavelength.
ε < 0: u=0 stable (disordered)
ε > 0: patterns form (stripes, spots, hexagons)
The competition between ε, cubic and quintic terms selects the pattern topology. Spiral and target patterns emerge from specific initial conditions. Related to Turing instability and Rayleigh-Bénard convection.