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Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract We present temporally first- and second-order accurate methods for the Swift–Hohenberg (SH) equation with quadratic–cubic nonlinearity. In order to handle the nonconvex, nonconcave term in the energy for the SH equation, we add an auxiliary term to make the combined term convex, which yields a convex–concave decomposition of the energy. As a result, the first- and second-order methods are unconditionally uniquely solvable and unconditionally stable with respect to the energy and pseudoenergy of the SH equation, respectively. And the Fourier spectral method is used for the spatial discretization. We present numerical examples showing the accuracy and energy stability of the proposed methods and the effect of the quadratic term in the SH equation on pattern formation. Highlights • We present temporally first- and second-order accurate methods for the SH equation with quadratic–cubic nonlinearity. • The unconditional unique solvability and unconditional stability of the proposed methods are analytically proven. • We numerically observe that pattern formation in the SH equation is mainly affected by its quadratic term. [ABSTRACT FROM AUTHOR]

Abstract The Swift–Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift–Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The solution is represented in the form of a series in the reproducing kernel space, and truncating this series representation we obtain the n -term approximate solution. In the first approach, we aim to explain how to construct a reproducing kernel method without using Gram-Schmidt orthogonalization, as orthogonalization is computationally expensive. This approach will therefore be most practical for obtaining numerical solutions. Gram-Schmidt orthogonalization is later applied in the second approach, despite the increased computational time, as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the Swift–Hohenberg equation. We demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values. [ABSTRACT FROM AUTHOR]

We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized- α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time. [ABSTRACT FROM AUTHOR]

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are L2 stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal L2 error estimate of O(hk+1) for polynomials of degree k for semi-discrete DG schemes, and the L2 error of O(hk+1+(Δt)2) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented. [ABSTRACT FROM AUTHOR]

This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical patternforming system, the Swift-Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross-Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed. [ABSTRACT FROM AUTHOR]

This paper presents results of the investigation of bifurcations of stationary solutions of the Swift-Hohenberg equation and dynamic descent to the points of minimal values of the functional of energy for this equation, obtained with the use of the modification of the Lyapunov-Schmidt variation method and some methods from the theory of singularities of smooth functions. Nonstationary case is investigated by the construction of paths of descent along the trajectories of the infinite-dimensional SH dynamical system from arbitrary initial states to points of the minimum energy. [ABSTRACT FROM AUTHOR]

We provide a multiple time scales analysis for the Swift–Hohenberg equation with delayed feedback via Pyragas control, with and without additive noise. An analysis of the pattern formation near onset indicates both the possibility of either standing waves (rolls) or traveling waves via Turing or Turing–Hopf bifurcations, respectively, depending on the product of the strength of the feedback and the length of the delay. The remainder of the paper is focused on Turing bifurcations, where the delay can drive the appearance of an additional time scale, intermediate to the usual slow and fast time scales observed in the modulation of rolls without delay. In the deterministic case, a Ginzburg–Landau-type modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein–Uhlenbeck-type equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy roll-type patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient. [ABSTRACT FROM AUTHOR]

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u ⁗ + q u ″ + f ( u ) = 0 , under various hypotheses on the parameter q and on the potential F ( t ) = ∫ 0 t f ( s ) d s , generally assumed to be bounded from below. We prove a non-existence result in the case q ≤ 0 and an existence result of periodic solution for: 1) almost every suitably small (depending on F ), positive values of q ; 2) all suitably large (depending on F ) values of q . Finally, we describe some conditions on F which ensure that some (or all) solutions u q to the equation satisfy ‖ u q ‖ ∞ → 0 , as q ↓ 0 . [ABSTRACT FROM AUTHOR]

The Swift–Hohenberg (SH) equation has been widely used as a model for the study of pattern formation. The SH equation is a fourth-order nonlinear partial differential equation and cannot generally be solved analytically. Therefore, computer simulations play an essential role in understanding of nonequilibrium processing. The aim of this paper is to present first- and second-order semi-analytical Fourier spectral methods as an accurate and efficient approach for solving the SH equation. The methods are based on the operator splitting method and are to split the SH equation into linear and nonlinear subequations. The linear and nonlinear subequations have closed-form solutions in the Fourier and physical spaces, respectively. The methods are simple to implement and computationally cheap to achieve high-order time accuracy. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods. [ABSTRACT FROM AUTHOR]

We study the effective temporal step size of convex splitting schemes for the Swift–Hohenberg (SH) equation, which models the pattern formation in various physical systems. The convex splitting scheme is one of the most well-known numerical approaches with an unconditional stability for solving a gradient flow. Its stability, solvability, and convergence have been actively studied; however, only a few studies have analyzed the time step re-scaling phenomenon for certain applications. In this paper, we present effective time step formulations for different convex splitting methods. Several numerical simulations are conducted to confirm the effective time step analysis. [ABSTRACT FROM AUTHOR]

The optical pulses generation in mode-locked lasers are described, in the case of complicated spectral filtering, by the cubic–quintic complex Swift-Hohenberg equation (CQSHE). Using the ansatz method, we studied the existence of the bright and dark chirped soliton solutions of the space-dependent coefficients CQSHE. The physical parameters of the chirped soliton solutions have been obtained as functions of the variable coefficients of the model equation. The parametric conditions for the existence of the chirped soliton solutions are presented. By the numerical evolution of the bright and dark solitons, we studied the influence of high order effects, dispersion and nonlinearity’s, on the propagation of the soliton solutions. [ABSTRACT FROM AUTHOR]

The generalization of the Swift–Hohenberg equation is studied. It is shown that the equation does not pass the Kovalevskaya test and does not possess the Painlevé property. Exact solutions of the generalized Swift–Hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained. The numerical algorithm which is based on the Crank–Nicolson–Adams–Bashforth scheme is developed. This algorithm is tested using the exact solutions. The selforganization processes described by the generalization of the Swift–Hohenberg equation are studied. [ABSTRACT FROM AUTHOR]

This paper is addressed to study the local exact controllability to the trajectories of the Swift–Hohenberg equation. We first establish a new global Carleman estimate for the fourth order parabolic operator, and we consider the exact controllability to the trajectories of the linearized system. Then, by Local inversion theorem, we show that the same result holds for the Swift–Hohenberg equation. [ABSTRACT FROM AUTHOR]

We propose a method for computation of stable and unstable sets associated to hyperbolic equilibria of nonautonomous ODEs and for computation of a specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain singular nonautonomous real Ginzburg-Landau type equation, which arises from the problem of formation of spots in the Swift-Hohenberg equation. [ABSTRACT FROM AUTHOR]

We propose a mathematical analysis of the Swift-Hohenberg equation arising from the phase field theory to model the transition from an unstable to a (meta)stable state. We also consider a recent generalization of the original equation, obtained by introducing an inertial term, to predict fast degrees of freedom in the system. We formulate and prove well-posedness results of the concerned models. Afterwards, we analyse the long-time behavior in terms of global and exponential attractors. Finally, by reading the inertial term as a singular perturbation of the Swift-Hohenberg equation, we construct a family of exponential attractors which is Höolder continuous with respect to the perturbative parameter of the system. [ABSTRACT FROM AUTHOR]

In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc $$\mathbb{D }$$ . Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of $$\mathbb{D }$$ also called H-planforms in reference with the 'planforms' introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of $$\mathbb{D }$$ and periodic in the 'transverse' direction. We highlight our theoretical results with a selection of numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we derive rigorously the amplitude equation, using the natural separation of time-scales near a change of stability, for the stochastic generalized Swift–Hohenberg equation with quadratic and cubic nonlinearity in this form du = - ( 1 + ∂ x 2 ) 2 u + ν ε u + γ u 2 - u 3 dt + σ ε dW , where W ( t ) is a Wiener process. For deterministic PDE it is known that the quadratic term generates an additional cubic term, which is unstable. We consider two cases depending on γ 2 . If γ 2 < 27 38 , then we have amplitude equation with cubic nonlinearities. In the other case γ 2 = 27 38 the cubic term in the amplitude equation vanishes. Therefore we consider larger solutions to obtain an amplitude equation with quintic nonlinearities. [ABSTRACT FROM AUTHOR]

We study the influence of boundary conditions on stationary, periodic patterns in one-dimensional systems. We show how a conceptual understanding of the structure of equilibria in large domains can be based on the characterization of boundary layers through displacement-strain curves. Most prominently, we distinguish wavenumber-selecting and phase-selecting boundary conditions and show how they impact the set of equilibria as the domain size tends to infinity. We illustrate the abstract concepts in the phase-diffusion and the Ginzburg-Landau approximations. We also show how to compute displacement-strain curves in more general systems such as the Swift-Hohenberg equation using continuation methods. [ABSTRACT FROM AUTHOR]

We study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations, which involve particular cases of the extended Fisher–Kolmogorov and Swift–Hohenberg equations. We address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution. The main result follows from an iterative method using the sequence of solutions obtained through the mountain-pass technique. In order to illustrate this method some numerical examples are shown. [ABSTRACT FROM AUTHOR]

This is a review article with a point of view. We summarize the long history of the subject and recent advances and suggest that almost all features of the architecture of shoot apical meristems can be captured by pattern-forming systems which model the biochemistry and biophysics of those regions on plants. [ABSTRACT FROM AUTHOR]

This paper is concerned with oscillatory behavior of a class of fourth-order delay dynamic equations on a time scale. In the general time scales case, four oscillation theorems are presented that can be used in cases where known results fail to apply. The results obtained can be applied to an equation which is referred to as Swift-Hohenberg delay equation on a time scale. These criteria improve a number of related contributions to the subject. Some illustrative examples are provided. [ABSTRACT FROM AUTHOR]

Abstract: The partially embedded submarine pipelines might buckle laterally at some segments under high pressure and high temperature (HP/HT) conditions. The buckling pattern localization introduces an extra level of analytical complexity when compared with the periodic buckling pattern. In the presented paper the lateral buckling pipeline is modeled by an axial compressive beam supported by lateral distributing nonlinear springs taking the soil berm effects in the horizontal plane. It is found that the model is governed by a time-independent Swift–Hohenberg equation. Based on John Burke and Edgar Knobloch’s work we conclude preliminarily that the equation will have localized solutions. Besides the qualitative conclusion, by AUTO 07P the localized solutions of the equation are studied in detail. The snakes-and-ladders structure of localized solutions explains the transition of buckling modes in theory. The range of the possible critical axial forces is found out. Meanwhile two critical axial force formulas corresponding to the range ends are presented. Finally a typical submarine pipeline is analyzed as an illustration. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the instability and the propagation properties of a class of reaction–diffusion equations of fourth order. Two examples are introduced, the extended Fisher Kolmogorov equation (EFK), and the Swift–Hohenberg equation (SH). Both have been studied before by related methods (see for example, Peletier and Rottschafer, 2004 [19]; Van Saarloos, 2003 [24]) but the analysis here will support the introduced linear mechanism in front selection. These two equations support a patterned front solutions, and the double eigenvalue mechanism is used to provide evidence for that and to determine a minimal front speed. [Copyright &y& Elsevier]

We study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of different orientations. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angles between the roll solutions. This extends prior work in [6] where the analysis was restricted to large angles, that is, weak bending near the grain boundary. The main new difficulty stems from possible interactions of the primary modes with other resonant modes. We generalize the normal form analysis in [6] and develop a singular perturbation approach to treat resonances. [ABSTRACT FROM AUTHOR]

Abstract: We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures. Building on work by Beck et al., the present work provides rigorous analytical results on the effects of perturbations to systems exhibiting snaking behavior. Starting with a reversible variational system possessing an additional symmetry, we elucidate the distinct effects of breaking symmetry and breaking variational structure, and characterize the resulting changes in both the bifurcation diagram and the solutions themselves. We show how to predict the branch reorganization and drift speeds induced by any particular given perturbative term, and illustrate our results via numerical continuation. We further demonstrate the utility of our methods in understanding the effects of particular perturbations breaking reversibility. Our approach yields an analytical explanation for previous numerical results on the effects of perturbations in the one-dimensional cubic–quintic Swift–Hohenberg model and allows us to make predictions on the effects of perturbations in more general settings, including planar systems. While our numerical results involve the Swift–Hohenberg model system, we emphasize the general applicability of the analytical results. [Copyright &y& Elsevier]