Nessa tese, o objetivo principal foi verificar a estabilidade de sistemas atômicos condensados, sujeitos a diferentes combinações lineares e não-lineares de redes ópticas bie tridimensionais, considerando algumas situações simétricas e assimétricas. Com esse objetivo, foram realizadas análises variacionais e simulações numéricas exatas da equação não-linear correspondente que descreve sistemas condensados de Bose-Einstein, tipo-Schrödinger, mais conhecida como equação de Gross-Pitaevskii. No caso bidimensional, com redes ópticas cruzadas, linear e não-linear, foi verificada a existência de estabilidade para certas regiões de parâmetros das interações. Observou-se que essa estabilidade desaparece ao se incluir uma terceira dimensão sem a presença de um potencial de confinamento. No caso tridimensional, considerando redes ópticas lineares e não-lineares cruzadas, a estabilidade só ocorre quando consideramos uma interação confinante na terceira dimensão, no caso, uma segunda rede óptica linear. Finalmente, espera-se que nossos resultados venham a ser úteis para estudos experimentais que vêm sendo feitos em laboratórios de átomos ultra-frios. / In this thesis, the main objective was the verification of stability of condensed atomic systems, subject to different combinations of linear and nonlinear bi- and tridimensional optical lattices , considering some symmetric and asymmetric situations. With this objective, were performed variational analyzes and numerical exact simulations of the nonlinear Schrödinger-type equation that describes Bose-Einstein condensate systems, better known as Gross-Pitaevskii equation. In two-dimensional case, with a crossed linear and nonlinear optical lattice, the stability was confirmed for certain parameter regions of the interactions. It was observed that the stability disappears when including a third dimension without the presence of a confinement potential. In the three dimensional case, considering crossed linear and nonlinear optical lattices, stability occurs only when considering an interaction confining the third dimension, in this case a second linear optical lattice. Finally, it is expected that our results will be useful for experimental studies which have been done in the laboratories of ultra-cold atoms. Keywords:

Cette thèse est dédiée à l'étude mathématique de l'approximation de champ-moyen des gaz de bosons. En physique quantique une telle approximation est vue comme la première approche permettant d'expliquer le comportement collectif apparaissant dans les systèmes quantiques à grand nombre de particules et illustre des phénomènes fondamentaux comme la condensation de Bose-Einstein et la superfluidité. Dans cette thèse, l'exactitude de l'approximation de champ-moyen est obtenue de manière générale comme seule conséquence de principes de symétries et de renormalisations d'échelles. Nous recouvrons l'essentiel des résultats déjà connus sur le sujet et de nouveaux sont prouvés, particulièrement pour les systèmes quantiques sur réseau, incluant le modèle de Bose-Hubbard. D'autre part, notre étude établit un lien entre les équations aux hiérarchies de Gross-Pitaevskii et de Hartree, issues des méthodes BBGKY de la physique statistique, et certaines équations de transport ou de Liouville dans des espaces de dimension infinie. Résultant de cela, les propriétés d'unicité pour de telles équations aux hiérarchies sont prouvées en toute généralité utilisant seulement les caractéristiques génériques de problèmes aux valeurs initiales liés à de telles équations. Egalement, de nouveaux résultats de caractères bien posés et un contre-exemple à l'unicité d'une hiérarchie de Gross-Pitaevskii sont prouvés. L’originalité de nos travaux réside dans l'utilisation d'équations de Liouville et de puissantes techniques de transport étendues à des espaces fonctionnels de dimension infinie et jointes aux mesures de Wigner, ainsi qu'à une approche utilisant les outils de la seconde quantification. Notre contribution peut être vue comme l'aboutissement d'idées initiées par Z. Ammari, F. Nier et Q. Liard autour de la théorie de champ-moyen. / This thesis is dedicated to the mathematical study of the mean-field approximation of Bose gases. In quantum physics such approximation is regarded as the primary approach explaining the collective behavior appearing in large quantum systems and reflecting fundamental phenomena as the Bose-Einstein condensation and superfluidity. In this thesis, the accuracy of the mean-field approximation is proved in full generality as a consequence only of scaling and symmetry principles. Essentially all the known results in the subject are recovered and new ones are proved specifically for quantum lattice systems including the Bose-Hubbard model. On the other hand, our study sets a bridge between the Gross-Pitaevskii and Hartree hierarchies related to the BBGKY method of statistical physics with certain transport or Liouville's equations in infinite dimensional spaces. As an outcome, the uniqueness property for these hierarchies is proved in full generality using only generic features of some related initial value problems. Again, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The originality in our works lies in the use of Liouville's equations and powerful transport techniques extended to infinite dimensional functional spaces together with Wigner probability measures and a second quantization approach. Our contributions can be regarded as the culmination of the ideas initiated by Z. Ammari, F. Nier and Q. Liard in the mean-field theory.

Ce travail est une contribution à l'étude des équations de Schrödinger non-linéaires (NLS) en dimension un d'espace. De telles équations interviennent notamment comme modèles dans plusieurs domaines de la physique mathématique, tels l'optique non-linéaire, la superfluidité, la supraconductivité et la condensation de Bose-Einstein.Cette thèse contient trois thèmes connexes inclus dans les chapitres 2, 3 et 4. Dans la première partie (chapitre 2), on s'intéresse à la construction des solutions en multi-solitons de l'équation de Gross-Pitaevskii (NLS défocalisante avec non-linéarité cubique), comme une superposition approximative des ondes progressives (solitons). Cette partie contient également une description détaillée des interactions entre les solitons. Ces résultats sont obtenus en exploitant l'intégrabilité de l'équation de Gross-Pitaevskii et son système de Marchenko associé.La deuxième partie (chapitre 4) clarifie les relations entre la formulation classique et la formulation dite hydrodynamique de l'équation de Gross-Pitaevskii. Cette dernière a un sens lorsque la solution ne s'annule jamais dans le domaine spatial. La dernière partie (chapitre 3) est consacrée à l'étude du problème de Cauchy d'une famille d'équations aux dérivées partielles quasi-linéaires qui généralise l'équation du flot par courbure binormal d'une courbe dans l'espace euclidien de dimension trois. Cette dernière est liée formellement à NLS par la transformation de Hasimoto. Dans notre généralisation, la vitesse d'un point de la courbe est toujours dirigée dans la direction du vecteur binormal, mais son amplitude peut dépendre de l'abscisse curviligne ainsi de la position dans l'espace. Notre approche pour prouver l'existence est le suivant: schéma semi-discret (discret en espace et continu en temps), obtention de bornes sur les problèmes discrets et argument par compacité. Un théorème de comparaison entraîne l'unicité. / This work is a contribution to the study of nonlinear Schrödinger equations (NLS) in the one-dimensional space. Such equations arise in many physical fields, including nonlinear optics and Bose-Einstein condensation. The thesis contains three connected themes included in chapters 2, 3 and 4. The first part (chapter 2) constructs multi-soliton solutions of the Gross-Pitaevskii (or defocussing NLS) equation, as an approximate superposition of traveling waves (solitons). This part contains also a detailed description of the interactions between solitons. These results are obtained by exploiting the integrability of the the Gross-Pitaevskii equation and its associated Marchenko system. The second part (chapter 4) clarifies the relations between the classical formulation and the so-called hydrodynamical formulation that only has a meaning when the solution does not vanish anywhere in the spatial domain The last part (chapter 3) of this thesis concerns existence and uniqueness results for a family of quasi-linear partial differential equations that generalize the equation of the binormal curvature flow for a curve in the three-dimensional space. The latter equation is in connection to the focussing cubic NLS by Hasimoto transformation. In our generalization, the velocity of a point on the curve is still directed along the binormal vector (so that in particular the length of the curve is preserved) but the magnitude of the speed is allowed to depend both on the curvilinear parameter and on the position in space. Existence is proven using spatial discretization together with some a priori bounds on the approximate solutions. Uniqueness follows from a comparison theorem.

No description available.

Ce mémoire présente plusieurs contributions quant à l'analyse mathématique de l'équation de Gross-Pitaevskii et du modèle de Bogoliubov-Dirac-Fock. Au sujet de l'équation de Gross-Pitaevskii, l'analyse commence par la construction variationnelle des ondes progressives minimisantes. La preuve de la stabilité orbitale du soliton noir en dimension un, et la description de la limite transsonique des ondes progressives minimisantes vers les états fondamentaux de l'équation de Kadomtsev-Petviashvili en dimension deux, viennent compléter cette construction. L'analyse s'achève par la dérivation rigoureuse du régime ondes longues vers l'équation de Korteweg-de Vries en dimension un. Quant au modèle de Bogoliubov-Dirac-Fock, il s'agit de construire les états fondamentaux du modèle réduit, puis de préciser le processus de renormalisation de leur charge, lequel autorise le calcul d'un développement asymptotique de la densité de charges du vide polarisé, qui est cohérent avec les développements perturbatifs de l'électrodynamique quantique.

Targeted Energy Transfer is a resonance phenomenon in coupled anharmonic oscillators. In this thesis we investigate if the concept of Targeted Energy Transfer is applicable to Bose-Einsteain condensates in optical lattices. The model used to describe Bose-Einstein condensates in optical lattices is based on the Gross-Pitaevskii equation. Targeted Energy Transfer in these systems would correspond to energy being transferred from one lattice site to another. We also try to expand the concept of Targeted Energy Transfer to a system consisting of three sites, where one of the sites are considered a perturbation to the system. We have concluded that it is possible to achieve Targeted Energy Transfer in a three-site system. The set-up of the system will in some of the cases studied lead to interesting properties, such as more energy being transferred to the acceptor site than what was initially localized on the donor site.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--Stellenbosch University, 2008. / The NLS is a universal equation of the class of nonlinear integrable systems. The aim of this thesis is to study the NLS numerically. More speci cally, an algorithm is developed to calculate its nonlinear spectrum. The nonlinear spectrum is then used as a diagnostic for numerical studies of the NLS. The spectrum consists of a discrete part, further subdivided into the main part, the auxiliary part, and the continuous spectrum. Two algorithms are developed for calculating the main spectrum. One is based on Floquet theory, rst implemented by Overman [12]. The other is a direct calculation of the eigenvalues by Herbst and Weideman [16]. These algorithms are combined through the marching squares algorithm to calculate the continuous spectrum. All ideas are illustrated by numerical examples.

Made available in DSpace on 2015-06-17T19:34:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-03-07. Added 1 bitstream(s) on 2015-06-18T12:48:41Z : No. of bitstreams: 1 000831658.pdf: 1830056 bytes, checksum: 3210c34fc6aebd29c87e24549ab48d45 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho teórico em Equações Diferenciais Parciais Elípticas, estudamos uma versão estacionária da equação de Schrödinger não-linear biharmônica. O objetivo principal versa sobre resultados de existência e concentração de soluções não-triviais, quando um parâmetro tende a zero. São utilizados métodos variacionais para estudar existência das soluções fracas não-triviais com hipóteses sobre o pontecial e sobre a não-linearidade / In this theoretical work in Elliptic Partial Di erential Equations, we study a stationary version of the biharmonic nonlinear Schrödinger equation. The main objective aims existence results and concentration of nontrivial solutions when a parameter tends to zero. Variational methods are used to study the existence of the weak nontrivial solutions under certain assumptions on the potential and the nonlinearity

Orientador: Marcos Tadeu de Oliveira Pimenta / Banca: Messias Meneguette Júnior / Banca: Michele de Oliveira Alves / Resumo: Neste trabalho teórico em Equações Diferenciais Parciais Elípticas, estudamos uma versão estacionária da equação de Schrödinger não-linear biharmônica. O objetivo principal versa sobre resultados de existência e concentração de soluções não-triviais, quando um parâmetro tende a zero. São utilizados métodos variacionais para estudar existência das soluções fracas não-triviais com hipóteses sobre o pontecial e sobre a não-linearidade / Abstract: In this theoretical work in Elliptic Partial Di erential Equations, we study a stationary version of the biharmonic nonlinear Schrödinger equation. The main objective aims existence results and concentration of nontrivial solutions when a parameter tends to zero. Variational methods are used to study the existence of the weak nontrivial solutions under certain assumptions on the potential and the nonlinearity / Mestre

Resumo: Investigamos as soluções estacionárias e dinâmicas da equação de Gross-Pitaevskii generalizada para sistemas atômicos com um potencial confinante e termos não conservativos associados à dissipação e à alimentação atômica, visando a descrição de condensados de Bose-Einstein. Consideramos os casos de comprimentos de espalhamento negativos (interações atrativas) e positivos (interações repulsivas) entre dois átomos. Verificamos como a variação dos parâmetros associados aos termos não conservativos pode produzir situações de instabilidade resultando no fenômeno conhecido como caos espaço temporal. Por outro lado, verificamos também quais combinações de parâmetros leva a soluções de equilíbrio, tipo solitônica. Nessa pesquisa, utilizando esse modelo de campo médio com uma parametrização conhecida, estudamos as propriedades de tais sistemas para alguns valores dos parâmetros não-conservativos, por meio de métodos numéricos e variacionais. / Abstract: We investigate the stationary and dynamical solutions of the Gross-Pitaevskii equation extended for atomic systems with confining potential in the presence of nonconservative terms associated to atomic dissipation and feeding, in order to describe Bose-Einstein Condensates. We considered the cases of negative (attractive interaction) and positive (repulsive interaction) two-body lenght. We verified how the variation of the parameters associated to those nonconservative terms could produce instabilities resulting in occurrence of spacetemporal chaos. In other hand, we looked for parameters combinations that give us stable solitonic-like solutions. In this research, by using the mean-field approach with a particular parameterization, we studied the properties of these systems for some values of the nonconservative parameters, by means of numerical and variational methods. / Orientador: Lauro Tomio / Coorientador: Victo dos Santos Filho / Banca: Gerson Francisco / Banca: Arnaldo Gammal / Mestre