 Swift–Hohenberg equation

The Swift–Hohenberg equation is a pattern-forming dissipative PDE with linear operator $r-(q_0^2+{\partial }_x^2{)}^2$. A spatial Fourier mode of wavenumber $k$ has growth rate $r-(q_0^2-k^2{)}^2$, selecting modes near $k=q_0$. A cubic-quintic version $w_t=[r-(1+{\partial }_x^2{)}^2]w+s{w}^3-w^5$ has destabilizing cubic feedback for $s>0$ and quintic saturation. Its weakly nonlinear envelope obeys a quintic real Ginzburg-Landau equation.

The Swift–Hohenberg equation is a partial differential equation that describes the evolution of certain patterns in nonlinear systems, particularly in the context of phase transitions and spatially extended systems. It is often used in the study of pattern formation in physical systems, such as fluid dynamics, chemical reactions, and biological systems.