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Chimera states consisting of domains of coherently and incoherently oscillating oscillators in a two-dimensional periodic array of nonlocally coupled phase oscillators are studied. In addition to the one-dimensional chimera states familiar from one spatial dimension, two-dimensional structures termed twisted chimera states and spiral wave chimera states are identified in simulations. The properties of many of these states, including stability, are determined using an evolution equation for a complex order parameter and are found to be in agreement with the simulations.

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter., 40 pages, 13 figures

The transition from subcritical to supercritical stationary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equation A(t) = μA + A(xx) + i(a(1)|A|(2)A(x) + a(2)A(2)A(x)*) + b|A|(2)|A - |A|(4)A, where A(x,t) represents the pattern amplitude and the coefficients μ, a(1), a(2), and b are real. The conditions for Eckhaus instability of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these evolve into spatially localized structures in the vicinity of a Maxwell point, while others resemble defect states. The results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality.

PACS number(s): 05.45.-a, 47.54.-r, 47.55.D, The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers, optical parametric oscillators, and photorefractive oscillators. We show that with real coefficients this equation exhibits two classes of localized states: localized in amplitude only or localized in both amplitude and phase. The latter are associated with phase-winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them. The localized states take the form of defects connecting phase-winding states with equal and opposite phase lag, and can be stable over a wide range of parameters. The formation of these defects leads to faceting of states with initially spatially uniform phase. Depending on parameters these facets may either coarsen indefinitely, as described by a Cahn-Hilliard equation, or the coarsening ceases leading to a frozen faceted structure., This work was funded by the Belgian Science Policy Office under Grant No. IAP-VI10. Support from the Research Foundation–Flanders (L.G.) and from the National Science Foundation under Grant No. DMS-0605238 (E.K.) is gratefully acknowledged. We thank J. Burke, S. Houghton, and U. Thiele for helpful discussions and a referee for bringing several references to our attention.

The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localized solutions organized in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility of the equation. We examine here the consequences of breaking spatial reversibility on the snakes-and-ladders structure. We find that the localized states now drift, and show that the snakes-and-ladders\ud structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques.

Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered.

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.

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. I. INTRODUCTION Ever since the observation that the subcritical complex Ginzburg-Landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states. The presence of these states has important consequences for other systems described by partial differential equations on the line since the Ginzburg-Landau equation describes the behavior of such systems near bifurcation from the trivial state of the system. Specifically, the complex Ginzburg-Landau equation describes the evolution of a long wavelength oscillatory instability, as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry. In contrast, near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real Ginzburg-Landau equation, and this equation possesses only unstable spatially localized states. It is of interest therefore to examine what happens to these unstable states at larger amplitude, where the real Ginzburg-Landau equation no longer provides an adequate description of the system. In this paper we show that the localized states can become stable at such amplitudes, and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions. We are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1:1 resonance, and use this theory to construct a large number of such states. The stability properties of these states are also determined, and the evolution of nonstationary localized states is studied by numerical integration in time. It is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation. The theory was originally developed in the context of water waves, where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ODE. The resulting localized states are called solitary waves, and in some cases turn out to be solitons. Kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area. Specifically, the ODE is viewed as a dynamical system in space, and localized states are sought as homoclinic orbits connecting the trivial state to itself. Whether such orbits are possible depends in part on the stability properties of the trivial state: eigenvalues with positive real part indicate that a nontrivial state can grow from x=, while eigenvalues with negative real part indicate that such a state may return, under appropriate conditions, back to the trivial state as x →. The spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system. In many cases, and in particular in the case considered here, the ODE is reversible. As a result the bifurcations that are encountered as a parameter is increased are nongeneric. In the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1:1 resonance. The unfolding of this resonance has been worked out in detail by Iooss and Peroueme 3, and can be used to understand the appearance of a variety of homoclinic orbits in this system, and hence of localized states with different spatial structure.

Direct numerical simulations of chevrons, blinking states, and repeated transients in binary fluid mixtures with a negative separation ratio heated from below are described. The calculations are performed in two-dimensional containers using realistic boundary conditions and the parameter values used in the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)]. Particular attention is paid to the multiplicity of states, and their dependence on the applied Rayleigh number and the aspect ratio of the container. Quantitative agreement with the experiments is obtained, and a mechanism explaining the origin and properties of the repeated transients observed in the experiments is proposed.

Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to three-dimensional patterns and are applied to two-species reaction-diffusion equations modeling the Turing instability.