The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as . We investigate the case , ν∈N^?, which encompasses the gravitational fluctuations, ν=4, and the electromagnetic waves, ν=2. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as , and we get global Lp estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L²−L^∞ estimates in suitable weighted Sobolev spaces.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

三维De Sitter与Anti-de Sitter空间中的等参曲面

A spacelike surface M in 3-dimensional de sitter space S13 or 3-dimensional anti-de Sitter space H13 is called isoparametric, if M has constant principal curvatures. A timelike surface is called isoparametric, if its minimal polynomial of the shape operator is constant. In this paper, we determine the spacelike isoparametric surfaces and the timelike isoparametric surfaces in S13 and H13.     

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.     

Nontrivial Soliton Scattering in 2+1 Anti-de Sitter Space

Using analytic methods, we present integrable solutions of the Bogomolny Yang–Mills–Higgs equations in 2+1 anti-de Sitter space. In particular, families of soliton solutions are constructed explicitly and their dynamics is investigated in some detail.     

Anti-de Sitte:空间H_1~3中的广义弹性曲线(英文)

In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing fields,we obtain existence for closed general-ized elastica fully immersed in Anti-de Sitter space H_1~3.     

三维 Anti-de Sitter空间中常平均曲率旋转曲面

本文主要构造三维 Anti-de Sitter空间中单参数族的常平均曲率旋转曲面.     

On the wave propagation in isotropic fractal media

In this paper, we explore the wave propagation phenomenon in three-dimensional (3D) isotropic fractal media through analytical and computational means. We present the governing scalar wave equation, perform its eigenvalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wave equation is derived makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspective, we consider the finite element method and derive the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method solves the resulting elastodynamic system and captures the transient response. Two solvers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are developed. They are validated in space and time, with particular problems considered on spherical shell domains. The first solver is elementary; it handles problems of purely radial dependence, effectively, 1D. However, the second one deals with general advanced 3D problems of arbitrary spatial dependence.     

On generalised thermoelastic wave propagation

Generalised thermoelasticity theories are employed to study one-dimensional disturbances in a half-space due to a thermal impulse on the boundary. Short time approximation of solutions are deduced and the exact discontinuities in the mechanical and thermal fields are analysed using the Laplace transform technique.     

On the wave propagation in isotropic fractal media

In this paper, we explore the wave propagation phenomenon in three-dimensional (3D) isotropic fractal media through analytical and computational means. We present the governing scalar wave equation, perform its eigenvalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wave equation is derived makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspective, we consider the finite element method and derive the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method solves the resulting elastodynamic system and captures the transient response. Two solvers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are developed. They are validated in space and time, with particular problems considered on spherical shell domains. The first solver is elementary; it handles problems of purely radial dependence, effectively, 1D. However, the second one deals with general advanced 3D problems of arbitrary spatial dependence.     

On electromagnetic wave propagation in fractional space

The vector Helmholtz equation in fractional space can describe the complex phenomenon of wave propagation in fractal media. With this view, a generalized Helmholtz equation for wave propagation in fractional space is established and its analytical solution is obtained. The special case for integer space is recovered and the results are in exact agreement with those obtained in the literature.     

The Propagator of a Massive Field of Spin s = 2 in the Anti-de Sitter Space

We construct the propagator of the massive tensor field of the second rank on the Euclidean continuation of the anti-de Sitter (AdS) space. We find the explicit expression for the propagator in the limit where the field takes values at the boundary of the AdS space. We show that the limiting expression yields the correct Green's function and two-point correlation function of the boundary conformal field theory, as predicted by the AdS/CFT correspondence hypothesis. We thus obtain one more piece of evidence in favor of the interpretation of operators of the boundary conformal field theory as certain limits of quantum fields propagating in the AdS space.     

On the longitudinal wave propagation in a viscoelastic cylinder

Résumé L'auteur étudie le calcul des déplacements d'un matériau visco-élastique linéaire dans un cylindre d'étendue finie qui sont produits par des oscillations longitudinales. Il montre qu'avec une approximation pour le cas des basses fréquences, on arrive au même résultat que celui obtenu de façon rigoureuse par un autre auteur. De plus, il montre que cette méthode peut être utilisée dans la region de hautes fréquences.     

On wave propagation in elastic tubes conducting rotational flows

The propagation of perturbations in liquid filled elastic tubes depends on the stream velocity of the basic flow. This phenomenon is currently analyzed with the method of the characteristics which relies upon a basic flow with a rectangular velocity profile. It seems that this one-dimensional flow approximation has not been convincingly validated, which justifies to consider other, more general velocity profiles.In the present analytical study the velocity profile is a quadratic function of the radial coordinate. Small amplitude perturbations are superposed on this inviscid, basic state in which the mean velocity is arbitrarily large. A normal mode analysis shows that the velocity profile and therefore the vorticity of the basic flow influence the more the phenomenon the larger is. For example, a parabolic profile allows countercurrent wave propagation regardless of.This questions the one-dimensional wave propagation theory in compliant tubes and, consequently, the interpretation of several physiological and medical problems mainly in the respiratory and cardio-vascular systems.

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Resumé La propagation de perturbations dans un tube élastique conduisant un écoulement fluide dépend de la vitesse de l'écoulement de base. Ce phénomène est habituellement étudié avec la méthode des caractéristiques, où l'on suppose que le profil de vitesse de l'écoulement est rectangulaire. Comme cette simplification ne semble pas avoir été bien validée, il paraît indiqué d'étudier l'impact d'autres profils.Dans la présente étude analytique, ce profil de vitesse est une fonction quadratique de la coordonée radiale. A cet écoulement non visqueux, dont la vitesse moyenne est arbitraire, l'on superpose des perturbations de faible amplitude. Une analyse linéarisée montre que le profil de vitesse et donc le rotationel de l'écoulement de base influencent d'autant plus ce phénomène d'ondes que est élévée.Ceci met en question la théorie uni-dimensionelle de la propagation d'ondes dans des tubes compliants et, par là-même, l'interpretation de divers problèmes physiologiques et médicaux, avant tout des systèmes respiratoires et cardio-vasculaires.

     

On the velocity of the Rayleigh wave propagation along curvilinear surfaces

To investigate the propagation of Rayleigh waves on curvilinear boundaries, wave propagation along cylindrical and spherical surfaces is considered. For elastic media with indicated boundaries, exact solutions of equations of elasticity theory are constructed and the asymptotics of Hankel and Legendre functions are used. On the basis of the results obtained, a conjecture is made concerning the dependence of the velocity of the Rayleigh wave on a small curvature of the route and on a small curvature in the perpendicular direction. Bibliography: 7 titles.     

On wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium

The paper examines the problem of wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details.     

On wave propagation and energy flux in thin transversely isotropic plates

A thin, unidirectionally reinforced, composite plate is considered and modelled as a transversely isotropic linear elastic plate. The structure is subjected to a oscillating load perpendicular to the plane of the plate at a central point. The resulting transverse (bending) waves are studied. Theoretical calculations using asymptotic expansions of the three-dimensional basic equations are presented. With the help of analytical solutions based on Fourier-transform techniques, calculated constant-phase lines, amplitude decay and energy flux are shown to be in good agreement with experiments performed on a thin unidirectionally reinforced plate made of carbon-fibre prepregs.

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Zusammenfassung Eine dúnne, unidirektionell verstárkte Verbundplatte wird als transversal isotropes Fláchentragwerk modelliert. Die Belastung besteht aus einer oszillierenden Last senkrecht zur Plattenebene in einem zentralen Punkt der Platte. Die entstehenden Biegewellen werden studiert. Theoretische Berechnungen beruhen auf asymptotischen Entwicklungen der dreidimensionalen Grundgleichungen. Analytische Lösungen, welche mit Hilfe von Fourier-Transformationsmethoden hergeleitet werden, erlauben die Berechnung von Kurven konstanter Phase, Amplitudenabnahme und Energiefluß. Die theoretischen Voraussagen stimmen mit experimentellen Beobachtungen und Messungen an einer kohlefaserverstärkten Platte gut überein.