Swift-Hohenberg equation

The Swift-Hohenberg equation is a partial differential equation noted for its pattern-forming behaviour. It takes the form

:$frac\{partial\; u\}\{partial\; t\}\; =\; r\; u\; -\; (1+\; abla^2)^2u\; +\; N(u)$

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 [J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convectiveinstability, Phys. Rev. A 15, 319-328 (1977)] , where it was derived from the equations for thermal convection.

The webpage of Michael Cross [ [http://www.cmp.caltech.edu/~mcc/Patterns/ Java applet demonstrations] ] contains some numerical integrators which demonstrate the behaviour of several Swift-Hohenberg-like systems.

References