PARAMETERS
Quick start:
Stripes
Hexagons
Near onset
Below onset
Pattern: —
Max |u|: —
Steps: 0
The Swift-Hohenberg equation:
∂u/∂t = εu − (∇²+q₀²)²u − u³
ε > 0: pattern-forming instability grows from noise. The operator (∇²+q₀²)² selects modes near wavenumber q₀.
Stripes form when initial conditions break hexagonal symmetry; hexagons need three-wave resonance. The cubic term u³ saturates growth and selects between patterns.
The preferred wavelength λ = 2π/q₀. Swift-Hohenberg is the normal form for the Turing instability and Rayleigh-Bénard convection near onset.