Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2010-08-23
Dawes J.H.P. 2010. Phil Trans. R. Soc. 368:3551--3565
Nonlinear Sciences
Pattern Formation and Solitons
14 pages, 6 figures. One contribution of 18 to a Triennial Issue `Visions of the future for the Royal Society's 350th annivers
Scientific paper
10.1098/rsta.2010.0057
Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific sub-class of such problems, where a pattern-forming, or `Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states, proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally.
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