= Pattern formation
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 $k=k_c\ne0$ 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 $k\simeq1$ near onset.
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