The emergence of a coherent structure for coherent structures: localized states in nonlinear systems

Abstract

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.

14 pages, 6 figures. One contribution of 18 to a Triennial Issue `Visions of the future for the Royal Society's 350th anniversary year'

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Cited by in corpus (18)

• Advances in Shell Buckling: Theory and Experiments
• From the granular Leidenfrost state to buoyancy-driven convection
• The Swift-Hohenberg equation with a nonlocal nonlinearity
• Variational approximations to homoclinic snaking in continuous and discrete systems
• Spatially Localized Structures in Lattice Dynamical Systems
• Variational approximations to homoclinic snaking
• Quantified 'shock sensitivity' above the Maxwell load
• Formation of localized states in dryland vegetation: Bifurcation structure and stability
• Variational approximation and the use of collective coordinates
• Localized patterns in planar bistable weakly coupled lattice systems
• Homoclinic snaking in the discrete Swift-Hohenberg equation
• Localised patterns in a generalised Swift--Hohenberg equation with a quartic marginal stability curve
• Snaking bifurcations of localized patterns on ring lattices
• Turbulent transition in a truncated one-dimensional model for shear flow
• Localized Patterns in Periodically Forced Systems: II. Patterns with Non-Zero Wavenumber
• Localised states in an extended Swift-Hohenberg equation
• Turbulent spots in channel flow: an experimental study Large-scale flow, inner structure and low order model
• A sequence of elastic patterns in a sheared bent sheet