Swift-Hohenberg Equation Research Articles

Multi-hump solutions with small oscillations at infinity for stationary Swift–Hohenberg equation

The paper considers the stationary Swift–Hohenberg equationcw−(∂x2+k02)2w−w3=0,where c > 0 is a constant, , and is a small parameter. In this case, the linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. It can be proved that the equation has homoclinic (or single hump) solutions approaching to periodic solutions as (called single-hump generalized homoclinic solutions). This paper provides the first rigorous proof of existence of homoclinic solutions with two humps which tend to periodic solutions at infinity (or two-hump generalized homoclinic solutions) by pasting two appropriate single-hump generalized homoclinic solutions together. The dynamical system approach is used to reformulate the problem into a classical dynamical system problem and then the solution is decomposed into a decaying part and an oscillatory part at positive infinity. By adjusting some free constants and modifying the single-hump generalized homoclinic solution near negative infinity, it is shown that the solution is reversible with respect to a point near negative infinity. Therefore, the translational invariant and reversibility properties of the system yield a two-hump generalized homoclinic solution. The method may be applied to prove the existence of 2k-hump solutions for any positive integer k.

Continuation and Bifurcation of Grain Boundaries in the Swift--Hohenberg Equation

We study grain boundaries between striped phases in the prototypical Swift-Hohenberg equation. We propose an analytical and numerical far- field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box with phase conditions. Using the spatially conserved quantities of the time-independent Swift-Hohenberg equation, we show that symmetric grain boundaries must select the marginally zig-zag stable stripes. We find that as the angle between the stripes is decreased, the symmetric grain boundary undergoes a parity-breaking pitchfork bifurcation where dislocations at the grain boundary split into disclination pairs. A plethora of asymmetric grain boundaries (with different angles of the far- field stripes either side of the boundary) is found and investigated. The energy of the grain boundaries is then mapped out. We find that when the angle between the stripes is greater than a critical angle, the symmetric grain boundary is energetically preferred while when the angle is less than the critical angle, the grain boundaries where stripes on one side are parallel to the interface are energetically preferred. Finally, we propose a classification of grain boundaries that allows us to predict various non-standard asymmetric grain boundaries.

PULLBACK ATTRACTORS FOR MODIFIED SWIFT-HOHENBERG EQUATION ON UNBOUNDED DOMAINS WITH NON-AUTONOMOUS DETERMINISTIC AND STOCHASTIC FORCING TERMS

In this paper, the existence and uniqueness of pullback attractors for the modified Swift-Hohenberg equation defined on Rn driven by both deterministic non-autonomous forcing and additive white noise are established. We first define a continuous cocycle for the equation in L2(Rn), and we prove the existence of pullback absorbing sets and the pullback asymptotic compactness of solutions when the equation with exponential growth of the external force. The long time behaviors are discussed to explain the corresponding physical phenomenon.

Bifurcation and final patterns of a modified Swift-Hohenberg equation

In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an \begin{document}$S^1$\end{document} -attractor as the control parameter \begin{document}$\alpha $\end{document} passes through a critical number \begin{document}$\hat{\alpha }$\end{document} . Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of finite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.

The solitary wave solution of the Swift-Hohenberg equation using He's semi inverse method

The solitary wave solution of the Swift-Hohenberg equation using He's semi inverse method

Nonlinear waves described by the generalized Swift-Hohenberg equation

We study the wave processes described by the generalized Swift-Hohenberg equation. We show that the traveling wave reduction of this equation does not pass the Kovalevskaya test. Some solitary wave solutions and kink solutions of the generalized Swift-Hohenberg equation are found. We use the pseudo-spectral algorithm to perform the numerical simulation of the wave processes described by the mixed boundary value problem for the generalized Swift-Hohenberg equation. This algorithm was tested on the obtained solutions. Some features of the nonlinear waves evolution described by the generalized Swift-Hohenberg equation are studied.

Formation of localized sand patterns downstream from a vertical cylinder under steady flows: Experimental and theoretical study.

The generation of localized, spatially periodic patterns on a sandy bottom is experimentally and theoretically studied. Tests are performed in a hydrodynamic flume where patterns are produced downstream from a vertical cylinder under a steady current. It is found that patterns appear as a result of a subcritical instability of the water-sand bottom interface. A dependence of the area shape occupied by the patterns on the flow velocity and the cylinder diameter is investigated. It is shown that the patterns' characteristics can be explained using the Swift-Hohenberg equation. Numerical simulations point out that for a correct description of the patterns, an additional term which takes into account the impact of vortices on the sandy bottom in the wake of a cylinder must be added in the Swift-Hohenberg equation.

Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion

We investigate the 1D Swift-Hohenberg equation with dispersion $$u_t+2u_{\xi\xi}-\sigma u_{\xi\xi\xi}+u_{\xi\xi\xi\xi}=\alpha u+\beta u^2-\gamma u^3,$$ where $\sigma, \alpha, \beta$ and $\gamma$ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term $-\sigma u_{\xi\xi\xi}$ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the first rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.

Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.

Stability on time-dependent domains: convective and dilution effects

We explore near-critical behavior of spatially extended systems on time-dependent spatial domains with convective and dilution effects due to domain flow. As a paradigm, we use the Swift–Hohenberg equation, which is the simplest nonlinear model with a non-zero critical wavenumber, to study dynamic pattern formation on time-dependent domains. A universal amplitude equation governing weakly nonlinear evolution of patterns on time-dependent domains is derived and proves to be a generalization of the standard Ginzburg–Landau equation. Its key solutions identified here demonstrate a substantial variety–spatially periodic states with a time-dependent wavenumber, steady spatially non-periodic states, and pulse-train solutions–in contrast to extended systems on time-fixed domains. The effects of domain flow, such as bifurcation delay due to domain growth and destabilization due to oscillatory domain flow, on the Eckhaus instability responsible for phase slips in spatially periodic states are analyzed with the help of both local and global stability analyses. A nonlinear phase equation describing the approach to a phase-slip event is derived. Detailed analysis of a phase slip using multiple time scale methods demonstrates different mechanisms governing the wavelength changing process at different stages.

Slow motion for the 1D Swift–Hohenberg equation

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus T. Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L1-close to a jump function v, remain close to v for large time. This can be achieved by regarding the equation as the L2-gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v.

Chirped soliton solutions of the cubic-quintic Swift-Hohenberg equation with space-dependent coefficients

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.

Dynamical behaviors of stochastic local Swift-Hohenberg equation on unbounded domain

In this paper, we first study the deterministic Swift-Hohenberg equation on a bounded domain. After obtaining some a priori estimates by the uniform Gronwall inequality, we prove the existence of an attractor by the Sobolev compact embeddings. Then, we consider the stochastic Swift-Hohenberg equation driven by additive noise on an unbounded domain and prove that the random dynamical system is asymptotically compact by uniform a priori estimates for the far-field values of the solution, which implies the existence of a random attractor for the random dynamical system associated with the stochastic Swift-Hohenberg equation.

On a Large Time-Stepping Method for the Swift-Hohenberg Equation

AbstractThe main purpose of this work is to contrast and analyze a large time-stepping numerical method for the Swift-Hohenberg (SH) equation. This model requires very large time simulation to reach steady state, so developing a large time step algorithm becomes necessary to improve the computational efficiency. In this paper, a semi-implicit Euler schemes in time is adopted. An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally. The stability property is proved rigorously based on an energy approach. Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.

Patterns formation in ferrofluids and solid dissolutions using stochastic models with dissipative dynamics

A functional with interactions short-range and long-range low coarse-grained approximation is proposed. This functional satisfies models with dissipative dynamics A, B and the stochastic Swift-Hohenberg equation. Furthermore, terms associated with multiplicative noise source are added in these models. These models are solved numerically using the method known as fast Fourier transform. Results of the spatio-temporal dynamic show similarity with respect to patterns behaviour in ferrofluids phases subject to external fields (magnetic, electric and temperature), as well as with the nucleation and growth phenomena present in some solid dissolutions. As a result of the multiplicative noise effect over the dynamic, some microstructures formed by changing solid phase and composed by binary alloys of Pb-Sn, Fe-C and Cu-Ni, as well as a NiAl-Cr(Mo) eutectic composite material. The model A for active-particles with a non-potential term in form of quadratic gradient explain the formation of nanostructured particles of silver phosphate. With these models is shown that the underlying mechanisms in the patterns formation in all these systems depends of: (a) dissipative dynamics; (b) the short-range and long-range interactions and (c) the appropiate combination of quadratic and multiplicative noise terms.

Localized structures and front propagation in systems with a conservation law

The properties of spatially localized structures in systems with a conservation law are reviewed and related to the properties of fronts in such systems. The theory is illustrated using the conserved Swift–Hohenberg equation, and the insights gained used to shed light on the process of crystallization from a melt and on two problems arising in hydrodynamics: convection in a rotating layer and convection in an imposed magnetic field. Spatially localized structures in systems with time-periodic forcing are also considered. Several open problems are highlighted.

Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation

This article deals with stationary localized solutions of the (2D) two-dimensional complex Swift-Hohenberg equation (CSHE). Our approach is based on the semi-analytical method of collective coordinate approach. According to the parameters of the equation and a suitable choice of ansatz, the stationary dissipative solitons of the 2D CSHE equation are mapped. This approach allows to describe the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics. Finally, the major impact of spectral filtering terms on the dynamic of the solitons is demonstrated.

Analytical and numerical solutions of the generalized dispersive Swift–Hohenberg equation

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.

Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation.As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift–Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space–time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.

Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects

Invariant solutions of shear flows have recently been extended from spatially periodic solutions in minimal flow units to spatially localized solutions on extended domains. One set of spanwise-localized solutions of plane Couette flow exhibits homoclinic snaking, a process by which steady-state solutions grow additional structure smoothly at their fronts when continued parametrically. Homoclinic snaking is well understood mathematically in the context of the one-dimensional Swift–Hohenberg equation. Consequently, the snaking solutions of plane Couette flow form a promising connection between the largely phenomenological study of laminar–turbulent patterns in viscous shear flows and the mathematically well-developed field of pattern-formation theory. In this paper we present a numerical study of the snaking solutions of plane Couette flow, generalizing beyond the fixed streamwise wavelength of previous studies. We find a number of new solution features, including bending, skewing and finite-size effects. We establish the parameter regions over which snaking occurs and show that the finite-size effects of the travelling wave solution are due to a coupling between its fronts and interior that results from its shift-reflect symmetry. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.