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Pattern formation is the development of spatial structure from an initially uniform state, often through loss of stability to perturbations with a preferred nonzero wavenumber. If a translation-invariant linearized equation has Fourier growth rate $\lambda (k)$, an unstable maximum near $k=k_c\ne 0$ initially amplifies spatial modes near that scale. Nonlinear interactions then determine whether the pattern saturates, drifts, oscillates or changes scale. The Swift–Hohenberg equation supplies a simple model with $\lambda (k)=r-(1-k^2{)}^2$, selecting $k\simeq 1$ near onset.

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.