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Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials and engineering sciences. The pursuit for theoretical descriptions of some among those physical problems led to the Swift–Hohenberg equation (SH3) which describes pattern selection in the vicinity of instabilities. A finite differences scheme, known as Stabilizing Correction (Christov and Pontes, 2002), developed to integrate the cubic Swift–Hohenberg equation in two dimensions, is reviewed and extended in the present paper. The original scheme features Generalized Dirichlet boundary conditions (GDBC), forcings with a spatial ramp of the control parameter, strict implementation of the associated Lyapunov functional, and second-order representation of all derivatives. We now extend these results by including periodic boundary conditions (PBC), forcings with Gaussian distributions of the control parameter and the quintic Swift–Hohenberg (SH35) model. The present scheme also features a strict implementation of the functional for all test cases. A code verification was accomplished, showing unconditional stability, along with second-order accuracy in both time and space. Test cases confirmed the monotonic decay of the Lyapunov functional and all numerical experiments exhibit the main physical features: highly nonlinear behavior, wavelength filter and competition between bulk and boundary effects. • We present a stable and easily adaptable splitting scheme for the cubic and quintic SH equation. • The scheme has second-order accuracy in time and space, strictly respecting the Lyapunov functional. • Periodic and Generalized Dirichlet boundary conditions are adopted for a uniform and structured grid. • We numerically study the effect of a nonuniform control parameter in both SH forms. [ABSTRACT FROM AUTHOR]

In this work, we propose a stabilized linear predictor–corrector scheme for the Swift–Hohenberg equation. More precisely, we apply a stabilized first-order scheme as the predictor and a stabilized second-order scheme as the corrector. We prove rigorously that our scheme satisfies the energy dissipation law and is second-order accurate. Numerical experiments are presented to show the accuracy and energy stability of our scheme. [ABSTRACT FROM AUTHOR]

Many systems of physical interest may be modelled by the bistable Swift–Hohenberg equation with cubic–quintic nonlinearity. We construct a two-parameter family of exact meromorphic solutions of the time-independent equation and use these to construct a one-parameter family of exact periodic solutions on the real line. These are of two types, differing in their symmetry properties, and are connected via an exact heteroclinic solution. We use these exact solutions as initial points for numerical continuation and show that some of these lie on secondary branches while others fall on isolas. The approach substantially enhances our understanding of the solution space of this equation. [ABSTRACT FROM AUTHOR]

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation. [ABSTRACT FROM AUTHOR]

Abstract: One control on the buckling of a layer (or layers) embedded in a weaker matrix is the reaction force exerted by the deforming matrix on the layer. If the system is linear and this force is a linear function of the layer deflection, as for linear elastic and viscous materials, the resulting buckles can be sinusoidal or periodic. However if the system is geometrically nonlinear, as in general non-coaxial deformations, or the matrix material is nonlinear, as for nonlinear elastic, non-Newtonian viscous and plastic materials, the buckling response may be localised so that individual packets of folds form; the resulting fold profile is not sinusoidal. These folds are called localised folds. Most natural folds are localised. One view is that irregularity derives solely from initial geometrical perturbations. We explore a different view where the irregular geometry results from a softening material or geometrical nonlinearity without initial perturbations. Localised folds form in a fundamentally different way than the Biot wavelength selection process; the concept of a dominant wavelength does not exist. Folds grow and collapse sequentially rather than grow simultaneously. We discuss the formation of localised folds with recent considerations of constitutive behaviour at geological strain rates for general three-dimensional deformations. [Copyright &y& Elsevier]

Abstract: In this article, the approximate solutions of the non-linear Swift Hohenberg equation with fractional time derivative in the presence of dispersive term have been obtained. The fractional derivative is described in Caputo sense. Time fractional nonlinear partial differential equations in the presence of dispersion and bifurcation parameters have been computed numerically to predict hydrodynamic fluctuations at convective instability for different particular cases and results are depicted through graphs. [Copyright &y& Elsevier]

We prove convergence of the stochastic exponential time differencing scheme for parabolic stochastic par-tial differential equations (SPDEs) with one-dimensional multiplicative noise. We examine convergence for fourth-order SPDEs and consider as an example the Swift-Hohenberg equation. After examining convergence, we present preliminary evidence of a shift in the deterministic pinning region [J. Burke and E. Knobloch, Localized states in the generalized Swift-Hohenberg equation, Phys. Rev. E. 73 (2006), pp. 056211-1-15; J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A 360 (2007), pp. 681-688; Y.-P. Ma, J. Burke, and E. Knobloch, Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1867-1883; S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1581-1592] with space-time white noise. [ABSTRACT FROM AUTHOR]

We study the existence of grain boundaries in the Swift–Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ordinary differential equations in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection. [ABSTRACT FROM PUBLISHER]

In this article, we study the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points. [ABSTRACT FROM AUTHOR]

Reversible variational partial differential equations such as the Swift–Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas. [ABSTRACT FROM PUBLISHER]

We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval (-ℓ, ℓ) with Dirichlet-Neumann boundary conditions, where ℓ > 0 is large and lies outside of some small neighborhoods of the points nπ and (n + 1/2)π, n ∈ N. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter O(ℓ-2)-close to its critical values when ℓ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on ℝ and admitting 2π-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the 2π spatially periodic case. [ABSTRACT FROM AUTHOR]

During the solidification of two-component solutions a two-phase mushy layer often forms consisting of solid dendritic crystals and solution in thermal equilibrium. Here, we extend previous weakly nonlinear analyses of convection in mushy layers to the derivation and study of a pattern equation by including a continuous spectrum of horizontal wave vectors in the development. The resulting equation is of the Swift-Hohenberg form with an additional quadratic term that destroys the up-down symmetry of the pattern as in other studies of non-Boussinesq convective pattern formation. In this case, the loss of symmetry is rooted in a non-Boussinesq dependence of the permeability on the solid-fraction of the mushy layer. We also study the motion of localized chimney structures that results from their interactions in a simplified one-dimensional approximation of the full pattern equation. [ABSTRACT FROM AUTHOR]

Abstract: In complicated bifurcation problems where more than one instability can arise at onset, reasonably sound derivations of the equations that govern the amplitudes of the nearly marginal modes have been developed when the spectrum of the modes is discrete. The basis of these derivations lies in the center manifold theorem of dynamical systems theory. But when the spectrum of the modes is continuous and we no longer have that theorem to fall back on, there is nevertheless an equation (the Swift–Hohenberg equation) that well describes the patterns seen in Rayleigh–Bénard convection. Indeed, several ‘derivations’ of the S–H equation have been offered and here we describe how to obtain the S–H equation using Bogoliubov’s method. We suggest that this procedure clarifies and simplifies (though it does not make rigorous) the derivation of the S–H equation. Looking ahead to the derivation of pattern equations for more complicated problems with continuous spectra, we also describe a diagrammatic procedure that, once mastered, is useful in performing the complicated perturbative developments that are needed in such derivations. Here we illustrate the proposed combination of the ideas of Bogoliubov and Feynman for the standard form of the Rayleigh–Bénard convection problem. The resulting pattern equation is nonlocal but it reduces without approximation to the 1-D Swift–Hohenberg equation in the case of 2-D convection. Like the S–H equation, the nonlocal version admits a Lyapunov functional and we briefly indicate its utility in pattern selection both for the Swift–Hohenberg equation and its nonlocal extension. We conclude by describing the kinds of problems for which we intend the combined method but reserve the exhibition of the required calculations for a future festschrift. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, bifurcation of a modified Swift–Hohenberg equation in two spatial dimension with periodic boundary conditions is considered, By using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The bifurcation structure of localized stationary radial patterns of the Swift–Hohenberg equation is explored when a continuous parameter is varied that corresponds to the underlying space dimension whenever is an integer. In particular, we investigate how 1D pulses and 2-pulses are connected to planar spots and rings when is increased from 1 to 2. We also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3. [Copyright &y& Elsevier]

This paper is concerned with the bifurcation of a complex Swift-Hohenberg equation. The attractor bifurcation of the complex Swift-Hohenberg equation on a onedimensional domain (0, L) is investigated. It is shown that the n-dimensional complex Swift-Hohenberg equation bifurcates from the trivial solution to an attractor under the Dirichlet boundary condition on a general domain and under a periodic boundary condition when the bifurcation parameter crosses some critical values. The stability property of the bifurcation attractor is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The problem of the Swift–Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress. [Copyright &y& Elsevier]

Abstract: The dynamics of vegetation is formulated in terms of the allometric and structural properties of plants. Within the framework of a general and yet parsimonious approach, we focus on the relationship between the morphology of individual plants and the spatial organization of vegetation populations. So far, in theoretical as well as in field studies, this relationship has received only scant attention. The results reported remedy to this shortcoming. They highlight the importance of the crown/root ratio and demonstrate that the allometric relationship between this ratio and plant development plays an essential part in all matters regarding ecosystems stability under conditions of limited soil (water) resources. This allometry determines the coordinates in parameter space of a critical point that controls the conditions in which the emergence of self-organized biomass distributions is possible. We have quantified this relationship in terms of parameters that are accessible by measurement of individual plant characteristics. It is further demonstrated that, close to criticality, the dynamics of plant populations is given by a variational Swift–Hohenberg equation. The evolution of vegetation in response to increasing aridity, the conditions of gapped pattern formation and the conditions under which desertification takes place are investigated more specifically. It is shown that desertification may occur either as a local desertification process that does not affect pattern morphology in the course of its unfolding or as a gap coarsening process after the emergence of a transitory, deeply gapped pattern regime. Our results amend the commonly held interpretation associating vegetation patterns with a Turing instability. They provide a more unified understanding of vegetation self-organization within the broad context of matter order–disorder transitions. [Copyright &y& Elsevier]

This paper is concerned with the asymptotic behavior of the solutions u(x,t) of the Swift–Hohenberg equation with quintic nonlinearity on a one-dimensional domain (0, L). With α and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches are also studied. Global bounds for the solutions u(x,t) are established and then the global attractor is investigated. Finally, with the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are closer, and their stabilities are analyzed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using Mawhin’s continuation theorem, we obtain existence results concerning -periodic solutions of fourth-order nonlinear differential equations of the type where , is continuous, -periodic in and is continuous, -periodic with mean value zero. Applications are given for the extended Fisher–Kolmogorov and Swift–Hohenberg equations with forcing term. [Copyright &y& Elsevier]

The development of spontaneous stationary equilibrium patterns induced by the injection of a laser pump field into a purely absorptive two-level atomic sodium vapour ring cavity is investigated by means of various weakly non-linear stability analyses applied to the appropriate governing evolution equation for this optical phenomenon. In the quasi-equilibrium limit for its atomic variables, the mathematical system modelling that phenomenon can be reduced to a single modified Swift–Hohenberg non-linear partial differential time-evolution equation describing the intracavity field on an unbounded 2D spatial domain. Diffraction of radiation can induce transverse patterns consisting of stripes, squares and hexagonal arrays of bright spots or honeycombs in an initially uniform plane-wave configuration. Then, these theoretical predictions are compared with both relevant experimental evidence and existing numerical simulations from some recent non-linear optical pattern formation studies. [ABSTRACT FROM PUBLISHER]

A method is described for calculating non-linear steady-state patterns in channels taking into account the effect of an end wall across the channel. The key feature is the determination of the phase shift of the non-linear periodic form distant from the end wall as a function of wavelength. This is found by analysing the solution close to the end wall, where Floquet theory is used to describe the departure of the solution from its periodic form and to locate the Eckhaus stability boundary. A restricted band of wavelengths is identified, within which solutions for the phase shift are found by numerical computation in the fully non-linear regime and by asymptotic analysis in the weakly non-linear regime. Results are presented here for the 2D Swift–Hohenberg equation but, in principle, the method can be applied to more general pattern-forming systems. Near onset, it is shown that for channel widths less than a certain critical value, the restricted band includes wavelengths shorter and longer than the critical wavelength, whereas for wider channels only shorter wavelengths are allowed. [ABSTRACT FROM PUBLISHER]

We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift- Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments. [ABSTRACT FROM AUTHOR]

We study the nature and stability of stationary solutions u(x) of the fourth order Swift–Hohenberg equation on a bounded domain (0, L) with boundary conditions u = 0 and u'' = 0 at x = 0 and x = L. It is well known that as L increases, the set of stationary solutions becomes increasingly complex. Numerical studies have exhibited two interesting types of structures in the bifurcation diagram for (L, u). In this paper we demonstrate through a center manifold analysis how these structures arise naturally near certain bifurcation points, and that there are no others. We also analyze their stability properties. [ABSTRACT FROM AUTHOR]

Abstract: The stability of planar fronts to transverse perturbations in bistable systems is studied using the Swift–Hohenberg model and an urban population model. Contiguous to the linear transverse instability that has been studied in earlier works, a parameter range is found where planar fronts are linearly stable but nonlinearly unstable; transverse perturbations beyond some critical size grow rather than decay. The nonlinear front instability is a result of the coexistence of stable planar fronts and stable large-amplitude patterns. While the linear transverse instability leads to labyrinthine patterns through fingering and tip splitting, the nonlinear instability often evolves to spatial mixtures of stripe patterns and irregular regions of the uniform states. [Copyright &y& Elsevier]

Abstract: We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed . We discuss a method that allows to derive bounds on the front velocity, and which, hence, can be used to prove for, among others, the Swift–Hohenberg equation, the extended Fisher–Kolmogorov equation and the cubic complex Ginzburg–Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift–Hohenberg equation are in full accord with our analytical predictions. [Copyright &y& Elsevier]

We study the large-time behaviour of solutions u(x,t) of the Swift–Hohenberg equation on a one-dimensional domain (0,L), focusing in particular on the role of the eigenvalue parameter α and the length L of the domain on the selection of the limiting profiles v. We show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles, and prove the existence of a collection Σn of disjoint intervals, which depend on the value of α, such that if L∈Σn then solutions all converge to the trivial solution v=0. We show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches. We establish global bounds for the solutions u(x,t) and identify the global attractor. Finally, we derive an estimate about the shape of the final state. [Copyright &y& Elsevier]

We study the existence of stationary solutions of a class of diffusion equations related to the so-called extended Fisher–Kolmogorov equation and the Swift–Hohenberg equation. We prove the existence of multitransition kinks and pulses. These solutions are obtained as local minima of the associated functional. For the Swift–Hohenberg equation, our result partially proves a numerical conjecture. [Copyright &y& Elsevier]

The nonlinear dynamics of a propagating flame front governed by a two-stage sequential chemical reaction is considered in the parameter range where the uniformly propagating front is unstable. We show that near the transition from the short wave to the long wave oscillatory instability the nonlinear dynamics is described by a Swift–Hohenberg equation with dominant dispersive term, coupled to an evolution equation for the zero mode associated with the translation symmetry of the propagating wave. The nonlinear dynamics described by this system of equations is studied both analytically and numerically. In the case of weak coupling between the two equations, we observe the spontaneous formation of spiral waves with rapidly moving cores, while strong coupling leads either to chaotic dynamics or to the formation of oscillons—spatially localized oscillating structures. [Copyright &y& Elsevier]

A method is described for calculating steady-state patterns in large-scale nonlinear systems, taking into account the effect of a lateral boundary and without the need for extensive numerical calculations. The key feature is the determination of the phase shift of the nonlinear periodic form distant from the boundary as a function of wavelength. This is found by analyzing the solution close to the boundary, where Floquet theory is used to describe the departure of the solution from its periodic form. For a restricted band of wavelengths lying within the Eckhaus boundary, dual solutions for the phase shift are found, one of which corresponds to an unstable state. Results are presented here for the one-dimensional Swift–Hohenberg equation in a semi-infinite domain but in principle the method can be applied to more general pattern-forming systems. The results are compared with the predictions of weakly nonlinear theory and with nonlinear computations on a large but finite domain. [Copyright &y& Elsevier]

Stationary even single-bump periodic solutions of the Swift-Hohenberg equation are analyzed. The coefficient k in the equation is found to be a critical parameter. It is proved if 0< k<1, there exist periodic solutions having the same energy as the constant solution u=0; if 1< k<3/2, there exist periodic solutions having the same energy as the stable states u=±√ k−1. The proof of the above results is based on a shooting technique, together with a linearization method and a scaling argument. [ABSTRACT FROM AUTHOR]

A multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth-order nonautonomous differential equations related either to the extended Fisher–Kolmogorov or to the Swift–Hohenberg equation is proved. Variational approach is used. Some numerical solutions are calculated via the finite element method. [Copyright &y& Elsevier]

Our aim in this article is to present a simplified form of the renormalization group (RG) method introduced by Chen, Goldenfeld, and Oono and to derive a rigorous study of the validity in time of the asymptotic solutions furnished by the RG method. We apply the renormalization group method to a slightly compressible fluid equation and to the Swift–Hohenberg equation. [ABSTRACT FROM AUTHOR]

We examine a variational problem from elastic stability theory: a thin elastic strut on an elastic foundation. The strut has infinite length, and its lateral deflection is represented by u : R → R. Deformation takes place under conditions of prescribed total shortening, leading to the variational problem (0.1) inf {½ ƒ u″2 + ƒ F(u): ½ ƒ u′2 = λ}. Solutions of this minimization problem solve the Euler-Lagrange equation (0.2) u″ ″ + pu″ + F′ (u) = 0, - ∞ < x < ∞. The foundation has a nonlinear stress-strain relationship F′, combining a destiffening character for small deformation with subsequent stiffening for large deformation. We prove that for every value of the shortening λ > 0 the minimization problem has at least one solution. In the limit λ → ∞ these solutions converge on bounded intervals to a periodic profile that is characterized by a related variational problem. We also examine the relationship with a bifurcation branch of solutions of (0.2), and show numerically that all minimizers of (0.1) lie on this branch This information provides an interesting insight into the structure of the solution set of (0.1). [ABSTRACT FROM AUTHOR]