Parameters
State
ε: 0.30
Time: 0
u range: [—, —]
Pattern: —
Steps/s: —
Theory
Swift-Hohenberg (1977):
Pattern selection:
• ε small: stripes (rolls)
• Larger ε: hexagons, squares
• Defect dynamics: dislocations and grain boundaries coarsen over time
Critical mode: exp(i·r) with |q|=1 grows for ε>0. Nonlinearity selects amplitude:
u_t = [ε−(1+∇²)²]u − u³
Linear instability at q=1 for ε>0.Pattern selection:
• ε small: stripes (rolls)
• Larger ε: hexagons, squares
• Defect dynamics: dislocations and grain boundaries coarsen over time
Critical mode: exp(i·r) with |q|=1 grows for ε>0. Nonlinearity selects amplitude:
|A|² = ε/3 (stripes)