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.