The Swift-Hohenberg equation 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.
References
^ J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319-328 (1977)
