Swift–Hohenberg equation
The Swift–Hohenberg equation (named after Jack B. Swift and Pierre Hohenberg) is a partial differential equation noted for its pattern-forming behaviour. It takes the form
where u = u(x, t) or u = u(x, y, t) is a scalar function defined on the line or the plane, r is a real bifurcation parameter, and N(u) is some smooth nonlinearity.
The equation is named after the authors of the paper,[1] where it was derived from the equations for thermal convection.
The webpage of Michael Cross[2] contains some numerical integrators which demonstrate the behaviour of several Swift–Hohenberg-like systems.
Another example where the equation appears is in the study of wrinkling morphology and pattern selection in curved elastic bilayer materials.[3][4]
The Swift–Hohenberg equation leads to the Ginzburg–Landau equation.
See also[edit]
- Dissipative soliton#Theoretical description
- Reaction–diffusion system
- Turing patterns
- Rayleigh–Bénard convection
References[edit]
- ^ J. Swift; P.C. Hohenberg (1977). "Hydrodynamic fluctuations at the convective instability". Phys. Rev. A. 15 (1): 319–328. Bibcode:1977PhRvA..15..319S. doi:10.1103/PhysRevA.15.319.
- ^ Java applet demonstrations
- ^ Stoop, Norbert; Lagrange, Romain; Terwagne, Denis; Reis, Pedro M.; Dunkel, Jörn (March 2015). "Curvature-induced symmetry breaking determines elastic surface patterns". Nature Materials. 14 (3): 337–342. doi:10.1038/nmat4202. ISSN 1476-1122.
- ^ Lewin, Sarah (8 April 2015). "A Grand Theory of Wrinkles". Quanta Magazine.