Pattern Localization in the Swift–Hohenberg Equation via Slowly Varying Spatial Heterogeneity

A new conservative Swift–Hohenberg equation and its mass conservative method

Annularly and radially phase-modulated spatiotemporal necklace-shaped patterns (SNPs) in the complex Ginzburg-Landau (CGL) and complex Swift-Hohenberg (CSH) equations are theoretically studied. It is shown that the annularly phase-modulated SNPs, with a small initial radius of the necklace and modulation parameters, can evolve into stable fundamental or vortex solitons. To the radially phase-modulated SNPs, the modulated "beads" on the necklace rapidly vanish under strong dissipation in transmission, which may have potential application for optical switching in signal processing. A prediction that the SNPs with large initial radii keep necklace-ring shapes upon propagation is demonstrated by use of balance equations for energy and momentum. Differences between both models for the evolution of solitons are revealed.

This article studies an internal type inverse problem that consists of recovering a nonlinear coefficient in the Swift-Hohenberg (SH) equation and establish the controllability to the trajectories of the corresponding control system. First, we establish a Carleman estimates for the linearized problem with Neumann type boundary conditions by internal observation. Applying this Carleman estimate and regularity results, we prove stability estimate of SH equation. Further, by using the fixed point arguments, we prove the null controllability of a linearized SH equation using Carleman estimate and observability inequality derived for adjoint equation. Finally, we extend it for the trajectory controllability of the nonlinear SH equation.

Global feedback control of pattern formation in a wide class of systems described by the Swift-Hohenberg (SH) equation is investigated theoretically, by means of stability analysis and numerical simulations. Two cases are considered: (i) feedback control of the competition between hexagon and roll patterns described by a supercritical SH equation, and (ii) the use of feedback control to suppress the blowup in a system described by a subcritical SH equation. In case (i), it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed (mixed-mode) hexagonal pattern. In case (ii), it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime. The effects of a delayed feedback are also investigated for both cases, and it is shown that delay can induce temporal oscillations as well as blowup.

Grain boundaries in two-dimensional traveling-wave patterns

The relaxation and hysteresis of a periodically forced Swift-Hohenberg (SH) equation as a phenomenological model for the magnetic domains of a garnet thin film in an oscillating magnetic field are studied.It is already known that the unforced SH equation settles down to a single type of spatial structure called a stripe pattern, and that the relaxation process yields a scaling law for the structure factor.Two types of temporally oscillating spatial structure consisting of stripe and polka-dot patterns have also been asymptotically observed in the case of a periodically forced SH equation.Relaxation scaling behaviors are studied for these two patterns.It is also shown for the forced case that a hysteresis is observed in the vicinity of the boundary between two different spatial patterns in the phase diagram.

The Swift–Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer (SIAM J Math Anal 55(3):2150–2185, 2023), we consider discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs.

We prove convergence of the stochastic exponential time differencing scheme for parabolic stochastic partial 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.

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

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

An energy stable method for the Swift–Hohenberg equation with quadratic–cubic nonlinearity

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.

We study the bifurcation analysis of the Swift–Hohenberg equation (SHE) with the odd-periodic condition as period parameter λ moves. Motivated by Peletier and Rottschäfer [Pattern selection of solutions of the Swift–Hohenberg equations. Phys D. 2004;194:95–126] and Peletier and Williams [Some canonical bifurcations in the Swift–Hohenberg equation. SIAM J Appl Dyn Syst. 2007;6:208–235], with the complete proof based on center manifold reduction, we show how the periodic SHE bifurcates from the trivial solution to an attractor when λ passes a critical number, and mainly when a gap collapsed to a point, and an overlapped interval emerges. Peletier and Williams provided the local behavior about nontrivial solutions of the SHE depending on critical lengths or the overlapped interval based on -norm. In this paper, by dropping the symmetric condition given in the paper by Peletier and Williams, we extend their results and find the explicit stationary solution of the SHE. We also present several numerical results explaining our results.

The Swift–Hohenberg equation accurately models the formation and evolution of patterns in a wide range of systems. However, in the field of fluid dynamics, two particular patterns arise during the Rayleigh-Benard convection, rolls and hexagons, and the formation of both has been simulated in this work. The Swift–Hohenberg (S–H) equation is a nonlinear partial differential equation of fourth order, and through an implicit finite differences method it has been numerically solved. A set of snapshots of the evolution of these patterns is shown.

In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.