Swift-Hohenberg Equation Research Articles

Learning PDE to Model Self-Organization of Matter.

A self-organization hydrodynamic process has recently been proposed to partially explain the formation of femtosecond laser-induced nanopatterns on Nickel, which have important applications in optics, microbiology, medicine, etc. Exploring laser pattern space is difficult, however, which simultaneously (i) motivates using machine learning (ML) to search for novel patterns and (ii) hinders it, because of the few data available from costly and time-consuming experiments. In this paper, we use ML to predict novel patterns by integrating partial physical knowledge in the form of the Swift-Hohenberg (SH) partial differential equation (PDE). To do so, we propose a framework to learn with few data, in the absence of initial conditions, by benefiting from background knowledge in the form of a PDE solver. We show that in the case of a self-organization process, a feature mapping exists in which initial conditions can safely be ignored and patterns can be described in terms of PDE parameters alone, which drastically simplifies the problem. In order to apply this framework, we develop a second-order pseudospectral solver of the SH equation which offers a good compromise between accuracy and speed. Our method allows us to predict new nanopatterns in good agreement with experimental data. Moreover, we show that pattern features are related, which imposes constraints on novel pattern design, and suggest an efficient procedure of acquiring experimental data iteratively to improve the generalization of the learned model. It also allows us to identify the limitations of the SH equation as a partial model and suggests an improvement to the physical model itself.

Stochastic attractor bifurcation of the one-dimensional Swift-Hohenberg equation with multiplicative noise

We study the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a one-dimensional domain (0,L). With α and the length of the domain L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and analyze the case when the bifurcation points nearly coincide. We show the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation.

Localized folding of thick layers

We describe the localized folding of thick layers embedded in a viscoelastic framework. Higher-order partial differential equations such as the Swift-Hohenberg equation are standard for modelling the folding process. Using a high-order shear theory, we modify the Swift-Hohenberg equation to describe the buckling of thick layers and consider the folded layer's viscoelastic behavior. The use of thick layers enables us to consider shear strains parallel to the layers during folding. Our model naturally captures the softening-stiffening behavior by including a non-linear viscoelastic description using a Winkler-type foundation. Next, we study the linear stability behavior of the system and derive the dispersion relations. Finally, we simulate this new model using a robust custom-built isogeometric analysis solver, which allows us to describe thick folded layers with localized folding. The numerical results show that the folding of an elastic layer produces periodic patterns while a viscoelastic layer deflects locally. When the horizontal forces are unequal, the periodic folds initiate in the direction of the smaller force and the localized deformations occur parallel to the larger force. Later, the nonlocal deformation occurs in the direction of a smaller force. Domes and basins or more linear ridges and valleys are formed according to the relative magnitudes of the applied forces. Domes and basins are the results of equal horizontal applied forces, and non-equal forces result in ridges and valleys.

An unconditionally energy-stable linear Crank-Nicolson scheme for the Swift-Hohenberg equation

In this work, we propose and analyze a stabilized linear Crank-Nicolson scheme for the Swift-Hohenberg equation. More precisely, we treat the nonlinear term explicitly while two second-order stabilization terms are added to improve the stability of the scheme. We show that our scheme satisfies the discrete energy dissipation. We prove rigorously that our scheme is second-order accurate in time. Moreover, we adopt a spectral-Galerkin approximation for the spacial variables and establish error estimates for the fully discrete scheme. Numerical experiments are presented to show the accuracy and energy stability of our scheme.

Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift-Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have 8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal patterns rotated by almost any angle $\alpha$ with respect to each other. Here, we revisit the Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.

Series Type Solution of Fuzzy Fractional Order Swift–Hohenberg Equation by Fuzzy Hybrid Sumudu Transform

There are some confusion and complexity in our everyday lives, as we live in an uncertain environment. In such type of environment, an accurate calculation of the data and finding a solution to a problem is not an easy job. So, fuzzy differential equations are the better tools to model problems in the fuzzy domain. Modeling the real-world phenomenon more accurately requires such operators. Therefore, we investigate the fractional-order Swift–Hohenberg equation in the fuzzy concept. We study this equation under the fuzzy Caputo fractional derivative. We use the fuzzy Sumudu transform to find out the semianalytical solution of the considered equation. To deal with the nonlinear term of the problem, we also use the Adomian decomposition method. To confirm the accuracy of the proposed procedure, we give two test problems. Lastly, we plot the numerical results for various fractional orders, and uncertainty belongs to [0, 1].

3D pattern formation from coupled Cahn-Hilliard and Swift-Hohenberg equations: Morphological phases transitions of polymers, bock and diblock copolymers

This work proposes a new mathematical model of a ternary mixture composed of coupled Cahn-Hilliard and Swift-Hohenberg equations solved in three dimensions (3D) through a pseudo-spectral implicit method (a numerical method of the fast Fourier transform). The 3D numerical solutions dynamic is analyzed through its structure factor and growth law to study its behavior and some observed phase transitions. These numerical solutions evolve into glassy and crystalline phases in the form of 3D patterns. The new model has a free energy functional that considers small monomers of polymer (or copolymeric) chains taken into the Edwards free energy, as well as solvent-monomers and solvent-stretch free energies whose dynamic is given by the Cahn-Hilliard and Allen-Cahn equations. Isotropic and anisotropic phases from polymers, block, and diblock copolymers blend such that these phases have a morphological diversity of classical and complex structures. As an application of the dynamics of the new ternary model, it has a diversity of morphologies which allows obtaining porous polymeric materials manufactured by a novel 3D printing technique: the mathematical design process and 3D printing assisted manufacturing (MDP-3DPAM). This method allows the proposal of a new technique for the design and creation of polymeric materials with controlled pore size distribution starting with computational modeling.

Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law

We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry x↦−x. This system exhibits a Turing instability and we study the dynamics close to the onset of this instability. First, we show that periodic traveling waves bifurcate from a homogeneous ground state. Second, fixing the bifurcation parameter close to the onset of instability, we construct modulating traveling fronts, which capture the process of pattern-formation by modeling the transition from the homogeneous ground state to the periodic traveling wave through an invading front. The existence proof is based on center manifold reduction to a finite-dimensional system. Here, the dimension of the center manifold depends on the relation between the spreading speed of the invading modulating front and the linear group velocities of the system. Due to the broken reflection symmetry, the coefficients in the reduced equation are genuinely complex. Therefore, the main challenge is the construction of persistent heteroclinic connections on the center manifold, which correspond to modulating traveling fronts in the full system.

Influence of higher-order nonlinear effects on optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser

The complex Swift-Hohenberg equation (CSHE) has been widely studied in recent years. It is a more real model in the mode-locked fiber laser with the saturable absorber due to the addition of the diffraction term compared with the complex Ginzburg-Landau equation. In this work, the dynamics process of the traditional soliton and M−type soliton pulses in the mode-locked fiber laser is demonstrated based on the CSHE model with higher-order nonlinear effects. The results show that with the increase of the small signal gain coefficient, the number of soliton molecules adds gradually. Under the same transmission conditions, the transmission of M−type solitons in the laser are more stable than that of single solitons. By adding the self-steepening effect, it can be found that the time-domain shift due to higher-order dispersion effects is compensated. The self-frequency shift effect caused by the Raman scattering can produce not only time domain shift, but also frequency domain shift. Moreover, the addition of higher-order diffraction term can describe the spectral response of multiple peaks, and makes the pulse spectrum show the asymmetric propagation in the transmission process. Finally, the increase of the length of the single-mode fiber on the right side of the gain fiber in the optical circuit will not only shift the center position of the output pulse backward, but also make the pulse energy show a ladder type downward trend.

Unconditionally energy stable C0-virtual element scheme for solving generalized Swift-Hohenberg equation

The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C0-virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems.

Stability and convergence analysis of adaptive BDF2 scheme for the Swift–Hohenberg equation

The design, analysis and numerical simulations of a stabilized variable time-stepping difference scheme, for the Swift–Hohenberg equation, are considered in this paper. The proposed time-stepping scheme is proved to preserve a discrete energy dissipation law. With the help of the new discrete orthogonal convolution kernels, the unique solvability and the unconditional energy stability of the numerical scheme are rigorously proved. In addition, the second-order L2 norm convergence both in time and in space, of the proposed scheme, is shown under the almost independent of the time-step ratios. To the best of our knowledge, this is the first time that the L2 norm convergence of the adaptive BDF2 method is achieved for the Swift–Hohenberg equation. Numerical examples are provided to illustrate our theoretical results and to show the computational efficiency of the numerical scheme.

Validated forward integration scheme for parabolic PDEs via Chebyshev series

In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introduce a zero finding problem F(a)=0 on a Banach algebra X of Fourier–Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part L=defDF(0) has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that L is an invertible linear operator on X, and we obtain explicit, rigorous and computable bounds for the operator norm ‖L−1‖B(X). These bounds are then used to verify the hypotheses of a Newton–Kantorovich type argument which shows that the (Newton-like) operator T(a)=defa−L−1F(a) is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fisher’s equation and the Swift–Hohenberg equation. We apply our approach to each of these models.

Invariant measures and statistical solutions for a nonautonomous nonlocal Swift–Hohenberg equation

This paper investigates a two-dimensional nonautonomous nonlocal Swift–Hohenberg equation with two kinds of kernels and studies the existence of invariant measures and statistical solutions, which are important research objects in the area of turbulence for fluid systems. The existence of weak solutions guarantees a norm-to-weak continuous process associated with the nonautonomous equation. We first prove the existence of the pullback attractor for the process via the pullback flattening. Then the unique existence of invariant measures is obtained by appropriate construction, so that the invariant measure is supported by this pullback attractor. This invariant measure is turned out to be exactly a statistical solution of the original nonlocal Swift–Hohenberg equation.

High order unconditionally energy stable RKDG schemes for the Swift–Hohenberg equation

We propose unconditionally energy stable Runge–Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows including the Swift–Hohenberg equation. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved for arbitrary time steps and spatial meshes. The method integrates a penalty free DG method for spatial discretization with a multi-stage algebraically stable RK method for temporal discretization by the energy quadratiztion (EQ) strategy. The resulting fully discrete DG method is proven to be unconditionally energy stable. By numerical tests on several benchmark problems we demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.

Fuzzy fractional estimates of Swift-Hohenberg model obtained using the Atangana-Baleanu fractional derivative operator

<abstract><p>Swift-Hohenberg equations are frequently used to model the biological, physical and chemical processes that lead to pattern generation, and they can realistically represent the findings. This study evaluates the Elzaki Adomian decomposition method (EADM), which integrates a semi-analytical approach using a novel hybridized fuzzy integral transform and the Adomian decomposition method. Moreover, we employ this strategy to address the fractional-order Swift-Hohenberg model (SHM) assuming g$ {\bf H} $-differentiability by utilizing different initial requirements. The Elzaki transform is used to illustrate certain characteristics of the fuzzy Atangana-Baleanu operator in the Caputo framework. Furthermore, we determined the generic framework and analytical solutions by successfully testing cases in the series form of the systems under consideration. Using the synthesized strategy, we construct the approximate outcomes of the SHM with visualizations of the initial value issues by incorporating the fuzzy factor $ \varpi\in[0, 1] $ which encompasses the varying fractional values. Finally, the EADM is predicted to be effective and precise in generating the analytical results for dynamical fuzzy fractional partial differential equations that emerge in scientific disciplines.</p></abstract>

Dissipative mechanism and global attractor for modified Swift-Hohenberg equation in $R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $R^{N}$ with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in $H^2(R^{N})$ in particular.

Numerical scheme for solving the nonuniformly forced cubic and quintic Swift–Hohenberg equations strictly respecting the Lyapunov functional

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.

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

Error estimate of a stabilized second-order linear predictor–corrector scheme for the Swift–Hohenberg equation

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.

Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.