Numerical analysis of a moving boundary problem in coastal hydrodynamics

A finite element model to tackle the moving boundary problem of wave run-up on moderately steep slopes is developed. The special aspects considered in this study are (1) the modification of shallow water equations to accommodate the effect of vertical accelerations and (2) the use of Lagrangian acceleration coupled with an element that adapts itself to the moving boundary closely. The pressure term in the one-dimensional momentum equation is derived using the Eulerian equation in the vertical direction. This takes care of the vertical accelerations which are significant during the motion of a wave on moderately steep slopes. The element near the boundary is allowed to change its dimension so that the fluid boundary is closely followed. Such a flexible element precludes the need for approximation of the variables with regard to the indefinite position of the boundary. This element is split into two when its dimension becomes unduly large compared to the unchanging elements. The need for such a splitting is shown by an examination of the entries in the global matrix. Results of water profile as a wave runs up a structure are given. A brief history of the work on similar problems is outlined.     

Existence and asymptotic stability for evolution problems on manifolds with damping and source terms

One considers the nonlinear evolution equation with source and damping terms

     

Nonlinear Internal Damping of Wave Equations with Variable Coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we study the energy decay rate when a nonlinear damping is applied on a suitable subrigion.     

Propagation of light-pulses at a negative group-velocity

This paper deals with the apparent superluminal propagation of electromagnetic pulses in a linear dispersive medium. One specifically examines the possibility that the pulse leaving the medium may be nearly identical to the incident one (low distortion) and in significant advance of it (strongly negative group-delays). Favourable conditions are obtained in a dilute medium where the required anomalous dispersion originates in an ensemble of narrow absorption or gain lines. Analytical expressions of the advancement of the pulse centre-of-gravity and of the lowest order distortion are established from the transfer-function of the medium. The experiments already achieved with arrangements involving a single absorption-line or a gain-doublet are analysed in detail and compared. The considerable difficulties to overcome in order to attain advancements comparable to the pulse width without important distortion are pointed out. New and promising schemes involving a narrow dip in a gain profile or absorption-doublets are proposed. Received 4 July 2002 / Received in final form 8 November 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: bruno.macke@univ-lille1.fr RID="b" ID="b"Unité Mixte de Recherche de l'Université et du CNRS (UMR 8523)     

Exact Internal Controllability of Maxwell's Equations

In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result. Accepted 30 September 1998     

Materials with a desired refraction coefficient can be made by embedding small particles

A method is proposed to create materials with a desired refraction coefficient, possibly negative one. The method consists of embedding into a given material small particles. Given n₀(x), the refraction coefficient of the original material in a bounded domain , and a desired refraction coefficient n(x), one calculates the number N(x) of small particles, to be embedded in D around a point xD per unit volume of D, in order that the resulting new material has refraction coefficient n(x).     

Stabilizing the intensity of a wave amplified by a beam of particles

The intensity of an electromagnetic wave interacting self-consistently with a beam of charged particles as in a free electron laser, displays large oscillations due to an aggregate of particles, called the macro-particle. In this article, we propose a strategy to stabilize the intensity by re-shaping the macro-particle. This strategy involves the study of the linear stability (using the residue method) of selected periodic orbits of a mean-field model. As parameters of an additional perturbation are varied, bifurcations occur in the system which have drastic effect on the modification of the self-consistent dynamics, and in particular, of the macro-particle. We show how to obtain an appropriate tuning of the parameters which is able to strongly decrease the oscillations of the intensity without reducing its mean-value.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

(2 + 1)D exotic Newton-Hooke symmetry, duality and projective phase

A particle system with a (2 + 1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and supercritical phases (describing 2D isotropic ordinary and exotic oscillators) are separated by the critical phase (one-mode oscillator), and are related by a duality transformation. In the flat limit, the system transforms into a free Galilean exotic particle on the noncommutative plane. The wave equations carrying projective representations of the exotic Newton-Hooke symmetry are constructed.     

A review of analytical methods, based on the wave approach, to compute partitions transmission loss

In this paper, various methods for calculating partition transparency are investigated. These methods are all based on the wave approach, yet they differ in the way of considering the incident field, partition leaf dimensions and the absorbing material, or absorbent, incorporated between the partition leaves. The method verification has been done through comparison with the experimental data available in the literature. Results are very convincing and the wave approach proves to be highly accurate. The basic wave approach is well suited for modeling of infinite single or double-leaf partitions, although the diffuse incident field requires spatial windowing to achieve agreement with experimental data. When porous material is incorporated inside double-leaf partitions, the model needs to be enhanced, based on the Biot theory, to ensure coincidence with experimental data. In the case of partitions, which cannot be considered infinite, spatial windowing is applied to the transparency to correct for the dimensional effect, especially in the low-frequency range. The final model turns out to be highly accurate, as long as spatial windowing is limited to the first coincidence frequency. The wave approach therefore proves to be a suitable method and calculation times are moreover acceptable.