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Hannover, University, Diss., 2003.

Hannover, University, Diss., 2003.

Konstanz, Universiẗat, Diss., 2002. / Dateiformat: tgz, Dateien im PDF-Format.

Universiẗat, Diss., 2003--Tübingen.

Dans cette thèse, nous étudions les équations d’Einstein dans le vide avec un champ de Killing de translation. En présence de cette symétrie, les équations d’Einstein dans le vide en dimension 3+1 peuvent s’écrire, dans le cas polarisé, comme un système d’équations d’Einstein couplées à un champ scalaire en dimension 2+1. Dans la première partie de cette thèse, nous étudions les équations de contraintes dans le cas asymptotiquement plat. Les équations de contraintes sont des équations de compatibilité qui doivent être satisfaites par les données initiales. Nous montrons l’existence de solutions pour des données assez petites, et introduisons un développement asymptotique faisant intervenir des quantités correspondant aux charges globales. Dans une deuxième partie nous montrons la stabilité de l’espace-temps de Minkowski avec un champ de Killing de translation, en temps exponentiellement grand par rapport à la petitesse de la donnée initiale. Nous travaillons dans les coordonnées d’onde généralisées. Nous introduisons une famille de métriques Ricci plates, et imposons le comportement asymptotique de nos solutions à l’extérieur du cône de lumière en choisissant un élément de cette famille de manière adéquate. Ce choix permet la convergence de nos solutions à l’intérieur du cône de lumière vers la solution de Minkowski. Dans la dernière partie de cette thèse nous étudions les équations de contraintes dans le cas compact hyperbolique. Nous montrons l’existence d’une équation limite associée aux équations de contraintes. / This thesis aim sat studying vacuum Einstein equations with a space-like Killing vector field. With this symmetry, 3+1 vacuum Einstein equations reduce, in the polarized case, to Einstein equations coupled to a scalar field in 2+ 1 dimensions. In the first part of this thesis, we study the constraint equations in the asymptotically flat case. The constraint equations correspond to computability conditions that the initial data must satisfy. We show the existence of solutions for small data, and we introduce an asymptotic expansion involving quantities which are the 2 dimensional equivalents for the global charges. In the second part, we show the stability of Minkowski space-time with a translation space-like Killing vector field in exponential time with respect to the smallness of initial data. We introduce a family of Ricci flat metrics, and we impose the asymptotic behaviour of our solutions in the exterior of the light cone by picking the right element in the family. This choice allows for the convergence to Minkowski solution in the interior of the light cone. In the last part of this thesis, we study the constraint equations in the compact hyperbolic case. We show the existence of a limit equation associated to the constraint equations.

Soliton solutions of Einstein's field equations for space–times with two non-null, commuting Killing Vectors are exact solutions obtained using the solution-generating techniques that resemble the well-known Inverse Scattering Methods that have been widely used m the solution of certain nonlinear PDE's such as Korteweg–de Vries, Sine–Gordon, non-linear Schrödinger. There exist two main soliton techniques in General Relativity. The Belinski–Zakharov technique allows for purely gravitational solutions. The Alekseev technique allows for solutions of the Einstein–Maxwell equations. In both techniques, solitons arise in connection with the poles of a certain so-called "dressing matrix".

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Física, Florianópolis, 2014. / Made available in DSpace on 2015-02-05T21:04:39Z (GMT). No. of bitstreams: 1 328309.pdf: 5901532 bytes, checksum: ddce1c80dc72c6bfa1c5aa724baeb7c6 (MD5) Previous issue date: 2014 / Obtivemos o controle sobre as dinâmicas coletivas de um duplo condensado de bósons através do uso de técnicas de impressão de fase sob uma abordagem de campo médio. Um condensado de bósons armadilhado em um potencial de duplo poço (uma junção Bose-Josephson) foi abordada por meio de simulações diretas da Equação de Gross-Pitaevskii. Com conservação de número de partículas e uma definição apropriada de diferença de fase fomos capazes de obter uma representação no espaço de fases das dinâmicas coletivas. O trânsito entre diferentes regimes dinâmicos coletivos foi gerado de forma controlada através do uso de impressões de fase. Alguns resultados secundários foram a avaliação numérica da validade do modelo de dois modos na descrição da dinâmica de Gross-Pitaevskii e uma aproximação não perturbativa semi-analítica das soluções com contrapartida linear da Equação de Gross-Pitaevskii unidimensional independente do tempo.<br> / Abstract : We had obtained the control over the collective dynamics of a double Bose-Einstein condensate by the use of phase imprinting techniques through a mean-field approach. A condensed Bose gas trapped in a double-well potential (a Bose-Josephson junction) is treated by direct simulations of the Gross-Pitaevskii equation. With number conserving and an appropriate definition of phase-difference we had been able to obtain a phase-space representation of the collective dynamics. The transit among the different collective regimes are generated in controlled way by the use of phase imprinting. Some secondary results were the numerical evaluation of the validity of the two-mode model to describe the Gross-Pitaevskii dynamics and a non-perturbative semi-analytic approximation of the solutions with linear counterpart of the time-independent unidimensional Gross-Pitaevskii equation.