The formation of steady patterns in one space dimension is generically
governed, at small amplitude, by the Ginzburg-Landau equation.
But in systems with a conserved quantity, there is a large-scale neutral
mode that must be included in the asymptotic analysis for pattern
formation near onset. The usual Ginzburg-Landau equation for the amplitude
of the pattern is then coupled to an equation for the large-scale mode.
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These amplitude equations show that for certain parameters all regular
periodic patterns are unstable. Beyond the stability boundary, there
exist stable stationary solutions in the form of spatially modulated
patterns or localised patterns. Many more exotic localised states are
found for patterns in two dimensions.
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Applications of the theory include convection in a magnetic field,
providing an understanding of localised states seen in numerical
simulations.