Jonathan H. P. Dawes

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking ∗

Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical “pattern-forming” instability and intertwine in a manner often referred to as “homoclinic snaking.” In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg–Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region...

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

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 subclass of such problems, in which 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 and 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.

Localized convection cells in the presence of a vertical magnetic field

Thermal convection in a horizontal fluid layer heated uniformly from below usually produces an array of convection cells of roughly equal amplitudes. In the presence of a vertical magnetic field, convection may instead occur in vigorous isolated cells separated by regions of strong magnetic field. An approximate model for two-dimensional solutions of this kind is constructed, using the limits of small magnetic diffusivity, large magnetic field strength and large thermal forcing. The approximate model captures the essential physics of these localized states, enables the determination of unstable localized solutions and indicates the approximate region of parameter space where such solutions exist. Comparisons with fully nonlinear numerical simulations are made and reveal a power-law scaling describing the location of the saddle-node bifurcation in which the localized states disappear.

Localized States in an Extended Swift–Hohenberg Equation

Recent work on the behavior of localized states in pattern-forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure; it is variational in time and conservative in space. In this paper we investigate an extended Swift-Hohenberg equation in which nonvariational and nonconservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. First we carry out the normal form analysis of the initial pattern-forming instability that leads to small-amplitude localized states. Next we examine the bifurcation structure of the large-amplitude localized states. Finally, we investigate the temporal stability of one-peak localized states. Throughout, we compare the localized states in the extended Swift-Hohenberg equation with the analogous solutions to the usual Swift-Hohenberg equation.

Modulated and Localized States in a Finite Domain

A subcritical pattern-forming (Turing) instability of a uniform state, in an infinite domain, produces two branches of spatially localized states that bifurcate from the pattern-forming instability along with a uniform spatially periodic pattern. In this paper we demonstrate that branches of localized states persist as strongly amplitude-modulated patterns in large, but finite, domains with periodic boundary conditions. Our analysis is carried out for a model Swift–Hohenberg equation with a cubic-quintic nonlinearity. If the domain size exceeds a critical value, modulated states appear in secondary bifurcations from the primary branch of spatially periodic solutions. Multiple-scales analysis indicates that these secondary bifurcations occur close to the primary instability and close to the saddle-node bifurcation on the spatially periodic solution branch. As the domain size increases, extra “turns” on the snaking curve arise through a repeating sequence of saddle-node bifurcations and mode interactions be...

Snaking and isolas of localised states in bistable discrete lattices

We consider localised states in a discrete bistable Allen-Cahn equation. This model equation combines bistability and local cell-to-cell coupling in the simplest possible way. The existence of stable localised states is made possible by pinning to the underlying lattice; they do not exist in the equivalent continuum equation. In particular we address the existence of ‘isolas’: closed curves of solutions in the bifurcation diagram. Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions. We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D.

Localized States in a Model of Pattern Formation in a Vertically Vibrated Layer

We consider a novel asymptotic limit of model equations proposed to describe the formation of localized states in a vertically vibrated layer of granular material or viscoelastic fluid. In physical terms, the asymptotic limit is motivated by experimental observations that localized states (“oscillons”) arise when regions of weak excitation are nevertheless able to expel material rapidly enough to reach a balance with diffusion. Mathematically, the limit enables a novel weakly nonlinear analysis to be performed which allows the local depth of the granular layer to vary by

On the seismic cycle seen as a relaxation oscillation

O(1)

Rapidly rotating thermal convection at low Prandtl number

amounts even when the pattern amplitude is small. The weakly nonlinear analysis and numerical computations provide a robust possible explanation of past experimental results.

Pattern selection in oscillatory rotating convection

An earthquake is commonly described as a stick-slip frictional instability occurring along preexisting crustal faults. The seismic cycle of earthquake recurrence is characterized by long periods of quasi-static evolution, which precede sudden slip events accompanied by elastic wave radiation: the earthquake. This succession of processes over two well-distinguished time-scales recalls the behavior of nonlinear relaxation oscillations. We explore this connection by studying, in the framework of rate-and-state friction, the sliding of two identical slabs of elastic solid driven in opposite directions with a constant relative velocity. Our first innovation is to establish that the motion of a spring–block system is an asymptotic mechanical analogue of the frictional sliding of a single interface from which elastic waves radiate. Due to wave reflection at the boundaries, the equivalent mass of the block M = k(h/c s )2/12 is not independent of the equivalent spring stiffness k, where h/2 denotes the slab thickness and c s is the shear wave speed. Considering a non-monotonic friction law, we show that the relaxation oscillation regime is reached when the characteristic time-scale of frictionless oscillations is much greater than the characteristic time of frictional memory effects: (M/k)1/2 ≫ L/V *. We combine a composite approximation of the stick-slip cycle and numerical studies to show that the interfacial relaxation oscillations result from the subtle interplay of the non-monotonic properties of the friction law driving the long stress build-up of the quasi-static phase, and the inertial control of the fast slip phase originating from the wave propagation. We discuss the geophysical consequences for earthquake mechanics, and connections between the rate-and-state and Coulomb models of friction.