= Pattern formation
<Pattern formation> is the development of spatial structure in a system. A common mechanism is the loss of <stability> of a uniform state to disturbances at a preferred nonzero <wavenumber>. In <Rayleigh-Bénard convection>, heating a fluid layer from below can destabilize the conducting state and produce organized <convection rolls>. In a <reaction-diffusion equation>, reaction and transport can instead select a spatial concentration pattern.
A linear calculation separates the possible scales. Write a small disturbance as a <Fourier mode> $e^{\lambda(k)t+ikx}$. Positive <real part> of $\lambda(k)$ means that mode initially grows, and the fastest growing <wavenumbers> often set the first visible scale. The <Swift–Hohenberg equation> has $\lambda(k)=r-(1-k^2)^2$, so at small positive $r$ it amplifies a band near $k=\pm1$.
Linear growth is only the onset calculation. A growing disturbance eventually becomes large enough for nonlinear terms to matter. Coupled <amplitude equations> can then describe competition between modes: one mode can suppress another, several can coexist, or the state can drift or oscillate. The resulting <planform> includes both the selected <wavevectors> and their relative <amplitudes> and <phases>.
A slowly varying <phase modulation> describes changes in a pattern's local position and orientation without immediately destroying its periodic structure. Its stability can differ from the stability of a uniform amplitude, as illustrated by the <zigzag instability> of parallel rolls.
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