This thesis is concerned with the existence and spectral stability of periodic waves in the fractional Korteweg-de Vries (KdV) equation and the fractional modified Korteweg-de Vries (mKdV) equation. We study the existence of periodic travelling waves using various tools such as Green's function for fractional Laplacian operator, Petviashvili fixed point method, and a new variational characterization in which the periodic waves in fractional KdV and fractional mKdV are realized as the constrained minimizers of the quadratic part of the energy functional subject to fixed L3 and L4 norm respectively. This new variational framework allows us to identify the existence region of periodic travelling waves and to derive the criterion for spectral stability of the periodic waves with respect to perturbations of the same period. / Thesis / Doctor of Philosophy (PhD)

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No / New representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane, and for the Green's function for sound propagation in the canyon formed by two infinitely high, parallel rigid or sound soft walls and an impedance ground surface. The infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a Laplace-type integral. A pole subtraction technique is used to remove poles in the integrand which lie near the path of integration, obtaining a smooth integrand, more suitable for numerical integration, and a specific numerical integration method is proposed. Numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types. It is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts, and for problems of scattering by periodic surfaces.

Este trabalho apresenta uma dedução das equações para a dinâmica de vórtices em superfícies utilizando argumentos físicos e balanço de momento, obtendo o resultado já conhecido devido a Boatto/Koiller e Hally. Na primeira parte, elaboramos uma releitura da contribuição de diversos pesquisadores incluindo, além dos já citados, o trabalho de Marchioro e Pulvirenti sobre a propriedade de localização para a equação de Euler e também a importante contribuição de Flucher e Gustafsson no que diz respeito à determinação da função de Green e função de Robin hidrodinâmicas em domínios do plano. Na segunda parte revisamos o problema da dinâmica de um traçador passivo induzida por um vórtice no disco unitário e estendemos para o caso com vorticidade de fundo constante. Por fim, analisamos a dinâmica de dois vórtices no toro plano, a qual reduz-se ao estudo da dinâmica do centro de vorticidade com hamiltoniana dada pela função de Green. É feita uma descrição das bifurcações das curvas de níveis desta hamiltoniana com respeito a variações do parâmetro modular. Mostramos que o campo hamiltoniano em questão é preservado por biholomorfismos e, portanto, o espaço dos parâmetros pode ser reduzido ao espaço de Moduli do toro plano. Mudanças dentro de uma mesma classe de equivalência por biholomorfismos podem alterar apenas a classe de homotopia das curvas de nível. / In this thesis the equations for the motion of vortices on Riemannian surfaces is studied. Using conservation of momentum and physical arguments, the classical equations of Hally and Boatto/Koiller are recovered. Then the localization result for the Euler\'s equation with flat metric (Marchioro and Pulvirenti) and the determination of the Green\'s and Robin\'s functions on plane domains are revisited in the context of Riemannian surfaces. On a second part of the thesis two examples are analyzed. At first the dynamics of a passive tracer in the unit disk on the flat plane with constant background vorticity. At second the dynamics of two vortices on flat tori. This last system is integrable. The dynamics is determined by the level sets of the Green\'s function which depends on the modular parameter of the torus. The full bifurcation diagram of the system as a function of the module parameter is determined.

Este trabalho apresenta uma dedução das equações para a dinâmica de vórtices em superfícies utilizando argumentos físicos e balanço de momento, obtendo o resultado já conhecido devido a Boatto/Koiller e Hally. Na primeira parte, elaboramos uma releitura da contribuição de diversos pesquisadores incluindo, além dos já citados, o trabalho de Marchioro e Pulvirenti sobre a propriedade de localização para a equação de Euler e também a importante contribuição de Flucher e Gustafsson no que diz respeito à determinação da função de Green e função de Robin hidrodinâmicas em domínios do plano. Na segunda parte revisamos o problema da dinâmica de um traçador passivo induzida por um vórtice no disco unitário e estendemos para o caso com vorticidade de fundo constante. Por fim, analisamos a dinâmica de dois vórtices no toro plano, a qual reduz-se ao estudo da dinâmica do centro de vorticidade com hamiltoniana dada pela função de Green. É feita uma descrição das bifurcações das curvas de níveis desta hamiltoniana com respeito a variações do parâmetro modular. Mostramos que o campo hamiltoniano em questão é preservado por biholomorfismos e, portanto, o espaço dos parâmetros pode ser reduzido ao espaço de Moduli do toro plano. Mudanças dentro de uma mesma classe de equivalência por biholomorfismos podem alterar apenas a classe de homotopia das curvas de nível. / In this thesis the equations for the motion of vortices on Riemannian surfaces is studied. Using conservation of momentum and physical arguments, the classical equations of Hally and Boatto/Koiller are recovered. Then the localization result for the Euler\'s equation with flat metric (Marchioro and Pulvirenti) and the determination of the Green\'s and Robin\'s functions on plane domains are revisited in the context of Riemannian surfaces. On a second part of the thesis two examples are analyzed. At first the dynamics of a passive tracer in the unit disk on the flat plane with constant background vorticity. At second the dynamics of two vortices on flat tori. This last system is integrable. The dynamics is determined by the level sets of the Green\'s function which depends on the modular parameter of the torus. The full bifurcation diagram of the system as a function of the module parameter is determined.

No description available.

Im Rahmen der vorliegenden Arbeit wird eine Tight-Binding-Formulierung der Korringa-Kohn-Rostoker-Greenschen-Funktionsmethode vorgestellt. Dabei werden mittels eines geeignet gewählten Referenzsystems abgeschirmte Strukturkonstanten konstruiert. Es werden die Vorteile und Grenzen dieser Transformation des Formalismus diskutiert. Es wird gezeigt, daß der numerische Aufwand zur erechnung der Elektronenstruktur von Systemen mit langgestreckter Elementarzelle linear mit der Systemgröße wächst. Damit ist eine Behandlung von Systemen mit 500 und mehr Atomen pro Elementarzelle möglich. Anhand von umfangreichen Testrechnungen kann demonstriert werden, daß das neue Verfahren bezüglich seiner Genauigkeit mit dem traditionellen KKR-Verfahren vergleichbar ist. Es werden Anwendungen zur Berechnung der Elektronenstruktur sowie zur Zwischenlagenaustauschkopplung von Co/Cu(100)-Vielfachschichten vorgestellt. / A newly developed ab initio tight-binding-formulation of the Korringa-Kohn-Rostoker-Green's function method for layered systems is presented. Screened structure constants are calculated by means of a repulsive reference system. Advantages and limits of this transformation of the formalism are discussed in detail. The numerical effort for self consistent electronic structure calculations of systems with a large prolonged supercell scales linearly with the system size. Systems with up to 500 atoms per unit cell can be treated easily. The accuracy of the new method is of the same order as the traditional KKR method. Applications to electronic structure calculations and magnetic interlayer exchange coupling in Co/Cu(100) multilayers are presented.

Let us consider the singularly perturbed model problem Lu := -epsilon laplace u-bu_x+cu = f with homogeneous Dirichlet boundary conditions on the unit-square (0,1)^2. Assuming that b > 0 is of order one, the small perturbation parameter 0 < epsilon << 1 causes boundary layers in the solution. In order to solve above problem numerically, it is beneficial to resolve these layers. On properly layer-adapted meshes we can apply finite element methods and observe convergence. We will consider standard Galerkin and stabilised FEM applied to above problem. Therein the polynomial order p will be usually greater then two, i.e. we will consider higher-order methods. Most of the analysis presented here is done in the standard energy norm. Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur? We will address this question by looking into a balanced norm. Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly. We will present estimates on the Green’s function associated with L, that can be used to derive pointwise error estimators.

L'objectif de cette thèse est de simuler l'histoire temporelle de la réponse d'un bâtiment de grande hauteur sous sollicitation sismique et de proposer des méthodologies simplifiées qui reproduisent correctement une telle réponse. Initialement, un modèle tridimensionnel par éléments finis est produit afin de valider sa fiabilité pour simuler le comportement réel du bâtiment pendant les mouvements du sol, enregistrés à l'aide d'accéléromètres. Il est proposé d'améliorer la précision du modèle numérique en imposant de multiples excitations, compte-tenu des effets de basculement et de la variabilité spatiale sur la sollicitation d'entrée. L'utilisation de fonctions de Green empiriques est proposée pour simuler la réponse sismique directement à partir d'enregistrements d'événements passés, sans avoir besoin de dessins de construction ni d'étalonnage des paramètres mécaniques. Une méthode de sommation stochastique, déjà utilisée pour prédire les mouvements du sol, est adoptée pour générer des signaux synthétiques à des hauteurs différentes du bâtiment, par extension du chemin de propagation des ondes du sol à la structure. Une représentation simplifiée du bâtiment comme une poutre homogène Timoshenko est proposée pour simuler la réponse sismique directement à partir des enregistrements des vibrations ambiantes. Des paramètres mécaniques équivalents sont identifiés à l'aide de l'interférométrie par déconvolution en termes de dispersion des ondes, de fréquences naturelles et de rapport de vitesse des ondes de cisaillement et de compression dans le milieu / The aim of this thesis is to simulate the time history response of a high rise building under seismic excitation and provide simplified methodologies that properly reproduce such response. Firstly, a detailed three-dimensional finite element model is produced to validate its reliability to simulate the real behavior of the building during ground motions, recorded using accelerometers. It is proposed to improve the accuracy of the numerical model by imposing multiple excitations, considering rocking effect and spatial variability on the input motion. The use of empirical Green's functions is proposed to simulate the seismic response directly from past event records, without the need of construction drawings and mechanical parameters calibration. A stochastic summation scheme, already used to predict ground motions, is adopted to generate synthetic signals at different heights of the building, extending the wave propagation path from the ground to the structure. A simplified representation of the building as a homogeneous Timoshenko beam is proposed to simulate the seismic response directly from ambient vibration records. Equivalent mechanical parameters are identified using deconvolution interferometry in terms of wave dispersion, natural frequencies and shear to compressional wave

We prove exponential decay for a fractional power of the Green's function for some correlated Anderson models using the fractional moment method.