On solving the potential equation

We study radial solutions to the generalized Swift-Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as r → ∞. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift-Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo-and heteroclinic orbits, and numerical simulation.     

Exact Meromorphic Stationary Solutions of the Cubic-Quintic Swift-Hohenberg Equation

In this paper, we study an ODE of the form b0u（4）＋b1u′′＋b2u＋b3u3＋b4u5=0, ′= d/dz＇ , which includes, as a special case, the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and Painleve analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].     

Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations

A multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth-order nonautonomous differential equations related either to the extended Fisher-Kolmogorov or to the Swift-Hohenberg equation is proved. Variational approach is used. Some numerical solutions are calculated via the finite element method.     

Stochastic Modulation Equations

Consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. Under appropriate scaling its solutions are approximated by a periodic wave modulated by the solutions to a stochastic Ginzburg-Landau equation. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

LOCALIZED PATTERNS OF THE SWIFT-HOHENBERG EQUATION WITH A DISSIPATIVE TERM

In this paper, the normal form analysis of quadratic-cubic Swift-Hohenberg equation with a dissipative term is investigated by using the multiple-scale method. In addition, we obtain Hamiltonian-Hopf bifurcations of two equilibria and homoclinic snaking bifurcations of one-peak and two-peak homoclinic solutions by numerical simulations.     

Homoclinic Solutions for Swift-Hohenberg and Suspension Bridge Type Equations

We establish the existence of homoclinic solutions for a class of fourth-order equations which includes the Swift-Hohenberg model and the suspension bridge equation. In the first case, the nonlinearity has three zeros, corresponding to a double-well potential, while in the second case the nonlinearity is asymptotically constant on one side. The Swift-Hohenberg model is a higher-order extension of the classical Fisher-Kolmogorov model. Its more complicated dynamics give rise to further possibilities of pattern formation. The suspension bridge equation was studied by Chen and McKenna (J. Differential Equations136 (1997), 325-355); we give a positive answer to an open question raised by the authors.     

Fronts,traveling fronts,and their stability in the generalized Swift-Hohenberg equation

The generalized Swift-Hohenberg equation with an additional quadratic term is studied. Time-stable localized stationary solutions of the pulse and front types are found. It is shown that stationary fronts give rise to traveling fronts, whose branches are also obtained. This study combines theoretical methods for dynamical systems (in particular, the theory of homo-and heteroclinic orbits) and numerical simulation.     

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 $R^{n}$ driven by both deterministic non-autonomous forcing and additive white noise are established. We first define a continuous cocycle for the equation in $L^{2}(R^{n})$, 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.     

Periodic and quasi-periodic solutions for the complex Swift-Hohenberg equation

In this paper, we consider the complex Swift-Hohenberg(CSH) equation $\frac{\partial u}{\partial t}=\lambda u-(\alpha+\mathrm{i}\beta)\l(1+\frac{\partial^2}{\partial x^2}\r)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the above equation.     

Swift-Hohenberg方程的打靶法

§ 1　IntroductionIn this paper we shall study the formation of spatially periodic patterns in extendedsystems described by Swift- Hohenberg equationut=ku - 1 +2x22 u - u3 ,k∈ R. (1.1)This equation was first proposed in 1976 by Swiftand Hohenberg[12 ] as a simple model forthe Rayleigh- B nard instability of roll waves.However,since then an effective m odel e-quation has been proved for a variety of system s in physics and mechanics.The Swift- Hohenberg equation has been studied a …     

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.     

Localized Roll-Solutions in the Ekman-Couette-System

In the Ekman-Couette-System, where the usual Couette-System is additionally rotated about its normal axis, localized single roll solutions have been known for some time. By varying different system parameters, a new localized solution emerging by saddle-node bifurcations from these known solutions has now been found. This new solution is of the same localized nature like the old solution but with an additional roll. Further bifurcations lead then to an increasing number of rolls, still localized. This behavior is kind of analogous to the so called ‘homoclinic snaking’ which has recently been investigated in conjunction with the Swift-Hohenberg equation and binary fluid convection. It might link the unstable localized single roll solutions with stable multi roll solutions or even with stable periodic roll solutions, which has to be shown yet. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stable Modulating Multipulse Solutions for Dissipative Systems with Resonant Spatially Periodic Forcing

Summary.    We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation, one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory. In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations. In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions for the class of systems under consideration. Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001     

一类广义Swift-Hohenberg模型方程的时间周期问题

本文研究一类广义Swift-Hohenberg模型方程的时间周期问题，证明问题周期广义解和周期古典解的存在性与唯一性．     

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.     

Oscillation of fourth-order delay dynamic equations

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.     

n维复Swift-Hohenberg方程的动态分叉

考虑复Swift-Hohenberg方程的分叉问题．首先对复Swift-Hohenberg方程在一维区域(0,L)上的吸引子分叉进行了考虑．而后给出了n维复Swift-Hohenberg方程,在一般区域上Dirichlet边界条件下和周期边界条件下,当参数λ穿过某些分叉点时从平凡解处分叉出吸引子,并对吸引子分叉的稳定性进行了分析．     

A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. This complements the result in [Smets, van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations 184 (2002) 78-96.] stating that for almost all there exists at least one solution. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions “close to” the computed approximations.     

Ground states for the higher-order dispersion managed NLS equation in the absence of average dispersion

The problem of existence of ground states in higher-order dispersion managed NLS equation is considered. The ground states are stationary solutions to dispersive equations with nonlocal nonlinearity which arise as averaging approximations in the context of strong dispersion management in optical communications. The main result of this note states that the averaged equation possesses ground state solutions in the practically and conceptually important case of the vanishing residual dispersions.     

Regularity of ground state solutions of dispersion managed nonlinear schrödinger equations

We consider the dispersion managed nonlinear Schrödinger equation (DMNLS) in the case of zero residual dispersion. Using dispersive properties of the equation and estimates in Bourgain spaces we show that the ground state solutions of DMNLS are smooth. The existence of smooth solutions in this case matches the well-known smoothness of the solutions in the case of nonzero residual dispersion. In the case x∈R² we prove that the corresponding minimization problem with zero residual dispersion has no solution.