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 . Positive real part of means that mode initially grows, and the fastest growing wavenumbers often set the first visible scale. The Swift–Hohenberg equation has , so at small positive it amplifies a band near .
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.
Pattern formation refers to the process by which ordered structures emerge in various natural systems. This phenomenon can be observed in a range of fields, including biology, physics, chemistry, and mathematics. The concept encompasses the mechanisms and principles that lead to the development of organized patterns from initially random states. In biological systems, pattern formation is prevalent in processes such as the development of animal coats (like spots or stripes), the arrangement of cells in tissues, and the morphogenesis of organisms.
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