# 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

${\displaystyle {\frac {\partial u}{\partial t}}=ru-(1+\nabla ^{2})^{2}u+N(u)}$

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.

## Applications

### Geometric Measure Theory

In 2009 Ruggero Gabbrielli[3] published a way to use the Swift-Hohenberg equation to find candidate solutions to the Kelvin Problem on minimal surfaces.[4][5]

## References

1. ^ J. Swift,P.C. Hohenberg (1977). "Hydrodynamic fluctuations at the convective instability". Phys. Rev. A. 15: 319–328. doi:10.1103/PhysRevA.15.319.CS1 maint: uses authors parameter (link)
2. ^ Java applet demonstrations