Fourth-order Evolution Equations Research Articles

Neighborhood filters and PDE’s

Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona–Malik equation, one of the first nonlinear PDE’s proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.

Equivalence group of a fourth-order evolution equation unifying various non-linear models

A fourth-order non-linear evolutionary partial differential equation containing several arbitrary functions of the dependent variable is considered. This equation arises as a generalization of various non-linear models describing a non-linear heat diffusion, the dynamics of thin liquid films, etc. Equivalence transformations give more flexibility to the unified model. We determine the generators of the equivalence group and use them for specifying certain types of arbitrary functions when the model equation has additional symmetries, and hence admits non-trivial group invariant solutions.

Stability and dynamics of transverse field structures in the optical parametric oscillator near resonance signal detuning

An order parameter equation is derived for an optical parametric oscillator near the resonance signal detuning limit in two dimensions. The parametric mixing between signal and pump fields with cavity diffraction leads to a quintic, fourth-order evolution equation of the Swift–Hohenberg type which supports the formation of cavity solitons, plane waves and periodic structures. The formation, interaction and stability of each of the nontrivial spatial structures is considered via extensive numerical simulations. Stable optical structures are created which can be used as the basis for a wide range of all-optical technologies.

Padded continued fraction absorbing boundary conditions for dispersive waves

Continued fraction absorbing boundary conditions (CFABCs) are new arbitrarily high-order local absorbing boundary conditions that are highly effective in simulating wave propagation in unbounded domains. The current versions of CFABCs are developed for the non-dispersive acoustic wave equation with convex polygonal computational domains. In this paper, the CFABCs are modified through augmentation of special padding elements, and are effective for absorbing evanescent waves occurring in dispersive wave problems while retaining their absorption properties for propagating waves. The padded CFABCs for dispersive wave equations result in fourth order evolution equations, which are solved using an efficient combination of Crank Nicholson method, Newmark time-stepping scheme, and operator splitting ideas. The effectiveness of the padded CFABCs and their implementation is illustrated through numerical examples with varying levels of dispersion.

Attractor for a conserved phase‐field system with hyperbolic heat conduction

AbstractWe consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.

On an Initial-Boundary Value Problem for a Fourth Order Composite Equation with Bilaplacian

An initial-boundary value problem for a fourth order evolution equation with the bilaplacian operator is considered. We prove the existence and uniqueness of a generalized solution. The proofs are mainly based on energy inequalities for the operator generated by the problem under consid- eration.

On the Geometric Flow of Kirchhoff Elastic Rods

Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Kirchhoff elastic rods in equilibrium.

Exponential attractors for a conserved phase-field system with memory

We consider a conserved phase-field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The model consists of a parabolic integrodifferential equation, coupled with a fourth-order evolution equation for the phase-field. This system, supplemented with suitable boundary conditions, can be interpreted as a dissipative dynamical system in a proper phase-space. In a previous joint work, the last two authors have proved the existence of a global attractor of finite fractal dimension. Here we show the existence of an exponential attractor, by means of the techniques we developed for nonconserved phase-field systems with memory.

Dissipativity of a conserved phase-field system with memory

We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels $a$ and $b$ account for relaxation effects in the energy constitutive equation and in the Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove that the system can be interpreted as a process. We show that it possesses an absorbing set, under three -thermodynamically consistent- conditions on the memory kernel $a$.

On a model for phase separation in binary alloys driven by mechanical effects

This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary) balance of linear momentum with a fourth-order evolution equation of Cahn–Hilliard type for the phase separation, and it is highly nonlinearly coupled. Existence and uniqueness results are shown.

Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves

A spatial two-dimensional version of the Zakharov equation describing the evolution of deep-water gravity waves is used to derive two fourth-order evolution equations, for the amplitudes of the surface elevation and of the velocity potential. The scaled form of the equations is presented.

Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order

The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt −a 1 u xx −a 2 u xxt =ϕ(u x ) x are proved.

The Willmore flow near spheres

The Willmore flow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of solutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere.

ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D'Alembert equation. Later on, Woinowsky-Krieger (Nash & Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends. Here a new equation for the small transverse vibrations of a simply supported beam is proposed. Such equation takes into account Kirchhoff's correction, as well as the correction for rotary inertia of the cross section of the beam and the influence of shearing strains, already present in the Timoshenko beam equation (cf. the Mindlin-Timoshenko equation for the plate). The model is inspired by a remark of Rayleigh, and by a joint paper with Panizzi & Paoli. It looks more complicated than the one proposed by Sapir & Reiss, but as a matter of fact it is easier to study, if a suitable change of variables is performed. The author proves the local well-posedness of the initial-boundary value problem in Sobolev spaces of order ≥ 2.5. The technique is abstract, i.e. the equation is rewritten as a fourth order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for the plate).

Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria

For a class of Hamiltonian systems in ℝ4 the set of homoclinic and heteroclinic orbits which connect saddle-focus equilibria is studied using a variational approach. The oscillatory properties of a saddle-focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle-foci. This variational approach does not require any assumptions on the intersections of stable and unstable manifolds, such as transversality. Moreover, these connections are shown to be local minimizers of an associated action functional. This result has applications to spatial pattern formation in a class of fourth-order bistable evolution equations.

Entry flow of fiber suspensions in a straight channel

The planar channel entry flow of a Newtown fluid containing neutrally buoyant, non-Brownian, slender particles is studied numerically. In particular, the effects of Reynolds number, Re, and a nondimensional suspension parameter, C, on the developing velocity and orientation fields are investigated. The governing orientation equations are solved along particle paths, whereas the flow kinematics is determined from an Eulerian viewpoint. The fourth-order orientation tensor, which characterizes the orientation structure, is obtained from the differential orientation evolution equations and also from the Lagrangian description of the orientation angles of a number of fictitious fibers in the orientation space. It is found that both the second- and fourth-order orientation evolution equations, if used with quadratic closure approximations, inaccurately predict an earlier alignment of fibers in the flow direction. Furthermore, for higher values of suspension parameter C, convergent results are not obtained by using such evolution equations. On the other hand, the results obtained by following a number of orientation angles indicate that the entry length increases linearly with C, and the effect of Reynolds number on the entry length becomes negligible for high C values.

On computation of solution curves for semilinear elliptic problems

We consider computation of solution curves for semilinear elliptic equations. In case solution is stable, we present an algorithm with monotone convergence, which is a considerable improvement of the corresponding schemes in [4] and [5]. For the unstable solutions, we show how to construct a fourth-order evolution equation, for which the same solution will be stable.

Compensation of the soliton self-frequency shift with phase-sensitive amplifiers

We analyze the effect of Raman scattering and higher-order dispersion on the propagation of short (1-ps) pulses in a nonlinear optical fiber in which the loss is balanced by a periodic chain of phase-sensitive, degenerate parametric amplifiers. The analysis shows that the Raman scattering does not induce a continuous frequency downshift on the pulse but only a small, finite frequency shift. Pulse propagation is governed by a nonlinear fourth-order evolution equation that admits stable solutions that propagate with a small, constant, inverse velocity in the group-volocity rest frame.

Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear

AbstractA fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

Long-distance pulse propagation in nonlinear optical fibers by using periodically spaced parametric amplifiers

We analyze pulse propagation in a nonlinear optical fiber in which linear loss in the fiber is balanced by a chain of periodically spaced, phase-sensitive, degenerate parametric amplifiers. Our analysis shows that no pulse evolution occurs over a soliton period owing to attenuation in the quadrature orthogonal to the amplified quadrature. Evidence is presented that indicates that stable pulse solutions exist on length scales much longer than the soliton period. These pulses are governed by a nonlinear fourth-order evolution equation, which describes the exponential decay of arbitrary initial pulses (within the stability regime) onto stable, steady-state, solitonlike pulses.