The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time.

The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.

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We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming PDE systems. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.The discovery of exact equilibrium and traveling-wave solutions to the full nonlinear Navier-Stokes equations has resulted in much recent progress in understanding the dynamics of linearly stable shear flows such as pipe, channel and plane Couette flow [1,2,3,4]. These exact solutions, together with their entangled stable and unstable manifolds, form a dynamical network that supports chaotic dynamics, so that turbulence can be understood as a walk among unstable solutions [5,6]. Moreover, specific exact solutions are found to be edge states [7], that is, solutions with codimension-1 stable manifolds that locally form the stability boundary between laminar and turbulent dynamics. Thus, exact solutions play a key role both in supporting turbulence and in guiding transition.This emerging dynamical systems viewpoint does not yet capture the full spatio-temporal dynamics of turbulent flows. One major limitation is that exact solutions have mostly been studied in small computational domains with periodic boundary conditions. The small periodic solutions cannot capture the localized structures typically observed in spatially extended flows. For example, pipe flows exhibit localized turbulent puffs. Similarly, in plane Couette flow (PCF), the flow between two parallel walls moving in opposite directions, localized perturbations trigger turbulent spots which then invade the surrounding laminar flow [8,9]. Even more regular long-wavelength spatial patterns such as turbulent stripes have been observed [10]. The known periodic exact solutions cannot capture this rich spatial structure, but they do suggest that localized solutions might be key in understanding the dynamics of spatially extended flows.Spatially localized states are common in a variety of driven dissipative systems. These are often found in a parameter regime of bistability (or at least coexistence) between a spatially uniform state and a spatially periodic pattern, such as occurs in a subcritical pattern forming instability. The localized state then resembles a slug of the pattern embedded in the uniform background. An early explanation of such states is due to Pomeau [11], who argued that a front between a spatially uniform and spatially periodic state, which might otherwise be expected to drift in time, can be stabilized over a finite parameter range by pinning to the spatial phase of the pattern. More recently, the details of this localization mechanism have been established for the subcritical Swift-Hohenberg equation (SHE) through a theory of spatial dynamics [12,13,14]. In one spatial dimension the time-inde...

We investigate the bifurcation structure of stationary localised patterns of the two-dimensional SwiftHohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localised roll, square and stripe patches that exhibit snaking and non-snaking behaviour on the same bifurcation branch. Some of these patterns snake between four saddle-node limits: recent analytical results predict then the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localised roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena we encounter.

Formation of spatially localized oscillations in parametrically driven systems is studied, focusing on the dominant 2:1 resonance tongue. Both damped and self-excited oscillatory media are considered. Near the primary subharmonic instability such systems are described by the forced complex Ginzburg-Landau equation. The technique of spatial dynamics is used to identify three basic types of coherent states described by this equation-small amplitude oscillons, large amplitude reciprocal oscillons resembling holes in an oscillating background, and fronts connecting two spatially homogeneous states oscillating out of phase. In many cases all three solution types are found in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The origin of this behavior can be traced to the formation of a heteroclinic cycle in space between the finite amplitude spatially homogeneous phase-locked oscillation and the zero state. The results provide an almost complete classification of the properties of spatially localized states within the one-dimensional forced complex Ginzburg-Landau equation as a function of the coefficients.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. This behavior is simplest to understand within the subcritical Swift-Hohenberg equation, but is also present in the subcritical regime of doubly diffusive convection driven by horizontal gradients. In systems that are unbounded in one spatial direction homoclinic snaking continues indefinitely as the localized structure grows to resemble a spatially periodic state of infinite extent. In finite domains or in periodic domains with finite spatial period the process must terminate. In this paper we show that the snaking branches in general turn over once the length of the localized state becomes comparable to the domain, and examine the factors that determine the location of the termination point or points, and their relation to the Eckhaus instability of the spatially periodic state.

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