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Spontaneous spatial patterns occur in nonlinear systems with spatial coupling, e.g., through diffraction or diffusion. Strong enough nonlinearity can induce breaking of spatial symmetry, such that a pattern becomes more stable than the original featureless state. Instances discussed are in nonlinear optics, but the phenomenons have a universal character and are the basis of spatial differentiation in nature, from crystals to clouds, from giraffe coats to galaxies. The basic theory and phenomenons of pattern formation are reviewed, with examples from experiments and simulations (mainly from optics). Patterns usually consist of repeated units, and such units may exist in isolation as localized structures. Such structures are akin to spatial solitons and are potentially useful in image and∕or information processing. The nature and properties of such structures are discussed and illustrated.
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