Swift-Hohenberg Equation Research Articles

Local control of globally competing patterns in coupled Swift-Hohenberg equations.

We present analytical and numerical investigations of two anti-symmetrically coupled 1D Swift-Hohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimension-two point of the Turing-wave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left- and right-traveling waves. In particular, these complex Ginzburg-Landau-type equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results.

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

We study the effect of domain growth on the orientation of striped phases in a Swift-Hohenberg equation. Domain growth is encoded in a step-like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern-forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern-forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill-posed, infinite-dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional-analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling-wave problem. We resolve the former difficulty using a farfield-core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot-strapping.

Stochastic Swift-Hohenberg Equation with Degenerate Linear Multiplicative Noise

We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear multiplicative noise acting on a finite set of modes of the dominant linear flow is introduced. Existence of a stochastic flow and a local stochastic invariant manifold for this stochastic form of SHE are both addressed in this work. We show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation, and obtain numerical evidence suggesting that this picture is a good approximation for the full system as well.

The phase structure of grain boundaries.

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 pattern-forming 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.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Wavenumber selection via spatial parameter jump.

The Swift-Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for x<0 and unstable for x>0. Using normal forms and spatial dynamics, we prove the existence of a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped pattern state. The wavenumbers of these stripes are contained in a narrow band whose width grows linearly with the size of the jump. This represents a severe restriction from the usual constant-parameter case, where the allowed band grows with the square root of the parameter. We corroborate our predictions using numerical continuation and illustrate implications on stability of growing patterns in direct simulations.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Detecting, anticipating, and predicting critical transitions in spatially extended systems.

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Connection between nonlinear energy optimization and instantons.

How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift-Hohenberg equation, which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances-minimal seeds-for transition to specific attractors from the ground state. Then, after generalizing the technique to consider multiple, equally spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton.

He’s homotopy perturbation method for solving time fractional Swift-Hohenberg equations

This paper find the most effective method to solve the time-fractional Swift-Hohenberg equation with cubicquintic non-linearity by combining the homotopy perturbation method and the fractional complex transform. The solution reveals some intermittent properties of thermal physics.

Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation

In this paper, we performed an investigation of the dissipative solitons of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE). Stationary to pulsating soliton bifurcation analysis of the 2D CSHE is displayed. The approach is based on the semi-analytical method of collective coordinate approach. This method is constructed on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. The reduced model helps to obtain approximately the boundaries between the stationary and pulsating solutions. We analyzed the dynamics and characteristics of the pulsating solitons. Then we obtained the bifurcation diagram for a definite range of the saturation of the Kerr nonlinearity values. This diagram reveals the effect of the saturation of the Kerr nonlinearity on the period pulsations. The results show that when the parameter of the saturation of the Kerr nonlinearity increases, one period pulsating soliton solution bifurcates to double period pulsations.

Homoclinic snaking in the discrete Swift-Hohenberg equation.

We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

An energy-stable generalized-α method for the Swift–Hohenberg equation

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. • We present a second-order energy-stable time-integration method for the Swift–Hohenberg equation that suppresses numerical instabilities. • Based on the generalized- α method, numerical dissipation is controlled via the spectral radius. • A detailed energy-stability proof and an estimate for the stabilization parameter are provided. • The proposed stabilization vanishes for sufficiently small time step sizes, recovering the original weak form of the equation. • Numerical results for a pattern-formation example are used to test the convergence and energy-stability. • We compare the primal and mixed formulation in terms of wall-clock time and approximation error.

Patterns and coherence resonance in the stochastic Swift–Hohenberg equation with Pyragas control: The Turing bifurcation case

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.

Existence and asymptotic behavior of nontrivial solutions to the Swift–Hohenberg equation

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u⁗+qu″+f(u)=0, under various hypotheses on the parameter q and on the potential F(t)=∫0tf(s)ds, 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 uq to the equation satisfy ‖uq‖∞→0, as q↓0.

The stochastic Swift–Hohenberg equation

In this paper, we will study the stochastic Swift–Hohenberg equation. The weak martingale solution, stationary martingale solution, invariant measures, mild solution, large deviation principle and random attractors for the stochastic Swift–Hohenberg equation will be considered.

A semi-analytical Fourier spectral method for the Swift–Hohenberg equation

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.

Random scalar fields and hyperuniformity

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystals and liquids. Hyperuniform systems have attracted recent attention because they are endowed with novel transport and optical properties. Recently, the hyperuniformity concept has been generalized to characterize two-phase media, scalar fields, and random vector fields. In this paper, we devise methods to explicitly construct hyperuniform scalar fields. Specifically, we analyze spatial patterns generated from Gaussian random fields, which have been used to model the microwave background radiation and heterogeneous materials, the Cahn-Hilliard equation for spinodal decomposition, and Swift-Hohenberg equations that have been used to model emergent pattern formation, including Rayleigh-Bénard convection. We show that the Gaussian random scalar fields can be constructed to be hyperuniform. We also numerically study the time evolution of spinodal decomposition patterns and demonstrate that they are hyperuniform in the scaling regime. Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg equation are effectively hyperuniform. We show that thresholding (level-cutting) a hyperuniform Gaussian random field to produce a two-phase random medium tends to destroy the hyperuniformity of the progenitor scalar field. We then propose guidelines to achieve effectively hyperuniform two-phase media derived from thresholded non-Gaussian fields. Our investigation paves the way for new research directions to characterize the large-structure spatial patterns that arise in physics, chemistry, biology, and ecology. Moreover, our theoretical results are expected to guide experimentalists to synthesize new classes of hyperuniform materials with novel physical properties via coarsening processes and using state-of-the-art techniques, such as stereolithography and 3D printing.

Proof of Quasipatterns for the Swift–Hohenberg Equation

This paper establishes the existence of quasipatterns solutions of the Swift–Hohenberg PDE. In a former approach we avoided the use of Nash–Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al related to the Nash–Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the freedom in the choice of parameterization of the bifurcating solution. Thanks to a transversality condition, the result gives only a bifurcating set, located in a small hornlike region centered on a curve, with the origin at the bifurcation point.

The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations

The collocation technique based on the radial basis functions (RBFs) method is simple and efficient for solving a wide area of problems. But the mentioned technique is poor for solving problems that have shock (advection problems) or the discontinuous initial condition. The local RBFs collocation technique is a meshless method based on the strong form. The use of local collocation RBFs method overcomes the mentioned important issue. In the current paper, based on the proposed idea in Wang (2015) [54], we consider a linear combination of shape functions of local radial basis functions collocation method and moving Kriging interpolation technique. For showing the efficiency of new technique, some multi-dimensional problems such as Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations have been chosen. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

Dynamic behaviors of a local modified stochastic Swift-Hohenberg equation with multiplicative noise

In this paper, we investigate a global random attractor for a stochastic local modified Swift-Hohenberg equation with multiplicative noise in Stratonovich sense. Through the Ornstein-Uhlenbeck (O-U) transformation, we obtain the random dynamical system associated with the stochastic local modified Swift-Hohenberg equation. Using the properties of the O-U process, we derive the specific uniform a priori estimates, using which we prove the existence of global random attractor for the corresponding random dynamical system.