= Swift–Hohenberg equation
{c}
{title2=$w_t=[r-(1+\partial_x^2)^2]w+sw^3-w^5$}
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 $w_t=[r-(1+\partial_x^2)^2]w+sw^3-w^5$ has destabilizing cubic feedback for $s>0$ and quintic saturation. Its weakly nonlinear envelope obeys a <quintic real Ginzburg-Landau equation>.
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