Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Benedito Leandro Neto.pdf: 688417 bytes, checksum: c1ac127d257e0a8d59d30de577413351 (MD5) Previous issue date: 2011-09-02 / In this work, we started with an historical study of Ricci Solitons showing that they, often, arise as a auto-similar solution for the Ricci flow. It was demonstrated, then, some basic concepts of Riemannian Geometry and a formal definition of a Ricci Solitons. To conclude the work, it was presented a study analysis of the [6] article, establishing , among other results, two theorems: the first one, an estimation for the potential function of a Gradient Shrinking Ricci Solitons, complete non-compact, and, the second one, an estimation for the volume of a Gradient Shrinking Ricci Solitons, complete non-compact. / Nesse trabalho, nós começamos com um levantamento histórico sobre os Ricci Sólitons, mostrando que, muitas vezes, eles surgem como solução auto-similar do fluxo de Ricci. Em seguida, introduzimos alguns conceitos básicos de geometria Riemanniana e definimos formalmente um Rici Sóliton. Concluimos o trabalho com um estudo aprofundado do artigo [6], do qual mostramos, dentre outros resultados, dois teoremas: uma estimativa para a função potencial de um Ricci Sóliton Gradiente Shrinking, completo e não-compacto e uma estimativa superior para o volume de um Ricci Sóliton Gradiente Shrinking, completo e não-compacto.

The topic of this thesis is about experimental study of phenomena which are associated with light propagation in nonlinear dielectric media. In the first part of this work, we study experimentally the snake instability of the bright soliton stripe of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. The instability is observed, through spectral measurements, on spatially extended femtosecond pulses propagating in a normally dispersive self-defocusing semiconductor planar waveguide. The second part of this thesis is about light propagation in nonlinear periodic media. We experimentally observe a stationary spatial gap (or Bragg) soliton in a periodic semiconductor planar waveguide. Based on the interference pattern of the soliton beam, we measure the power parameter of the soliton which is related to the position of the spatial spectrum in the linear band gap. <p><p><p>Cette thèse de doctorat a pour sujet l’étude expérimentale de phénomènes associés à la propagation de la lumière dans les milieux diélectriques non linéaires. La première partie porte sur la démonstration expérimentale de l’instabilité serpentine d’une bande solitonique dans un système décrit par une équation de Schrödinger non linéaire à (2+1)-dimensions. L’instabilité est observée sur base de mesures du spectre spatial ainsi que du profil spatio-fréquentiel d’une impulsion femtoseconde après propagation dans un guide plan semi-conducteur qui présente une dispersion normale et une non-linéarité défocalisante. Le second thème abordé concerne la propagation de la lumière dans les milieux non linéaires périodiques. Les expériences réalisées ont montré l’existence du soliton de Bragg spatial stationnaire sous forme de faisceaux se propageant dans des guides plans semi-conducteurs périodiquement gravés. Sur base du profil de la distribution modale en intensité du faisceau soliton, il a été possible de mesurer le paramètre de puissance du soliton de Bragg qui détermine la position du spectre spatial dans la bande interdite linéaire. <p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

We investigate classically (iso)spinning topological soliton solutions in (2+1)- and (3+1)-dimensional models; more explicitly isospinning lump solutions in (2+1) dimensions, Skyrme solitons in (2+1) and (3+1) dimensions and Hopf soliton solutions in (3 +1) dimensions. For example, such soliton types can be used to describe quasiparticle excitations in ferromagnetic quantum Hall systems, can model spin and isospin states of nuclei and may be candidates to model glueball configurations in QCD.Unlike previous work, we do not impose any spatial symmetries on the isospinning soliton configurations and we explicitly allow the isospinning solitons to deform and break the symmetries of the static configurations. It turns out that soliton deformations clearly cannot be ignored. Depending on the topological model under investigation they can give rise to new types of instabilities, can result in new solution types which are unstable for vanishing isospin, can rearrange the spectrum of minimal energy solutions and can allow for transitions between different minimal-energy solutions in a given topological sector. Evidently, our numerical results on classically isospinning, arbitrarily deforming solitons are relevant for the quantization of classical soliton solutions.

This project studied two different ways of imposing boundary conditions weakly with the finite difference summation-by-parts (SBP) operators. These operators were combined with the boundary handling methods of simultaneous-approximation-terms (SAT) and the Projection to impose homogeneous Neumann and Dirichlet boundary conditions. The convergence rate of both methods was analyzed for different boundary conditions for the one-dimensional (1D) Schrödinger equation, without potential, which resulted in both methods performing similarly. A multi-block discretization was then implemented and different combinations of SBP-SAT and SBP-Projection were applied to impose inner boundary conditions of continuity between the blocks. A convergence study of the different methods of imposing the inner BC:s was conducted for the 1D Schrödinger equation without potential. The resulting convergence was the same for all methods and it was concluded that they performed similarly. Methods involving SBP-Projection had the slight advantage of faster computation time. Finally, the 1D Gross-Pitaevskii equation (GPE) and the 1D Schrödinger equation were analyzed with a step potential. The waves propagating towards the potential barrier were in both cases partially transmitted and partially reflected. The waves simulated with the Schrödinger equation dispersed, while the solitons simulated with the GPE kept their shape due to the equations reinforcing non-linear term. The bright soliton was partly transmitted and partly reflected. The dark soliton was either totally reflected or totally transmitted.

The work deals with dispersive effects of single mode fibers. We learn about the chromatic dispersion, the main attention is paid to the Polarization Mode Dispersion. We clarify the root cause origin of the Polarization Mode Dispersion describing parameters and its effect on transmitted data. Further on, the works dedicates to measure methods of the Polarization Mode Dispersion which is the interferometric method or the POTDR method. We explain the ways of the PMD removal at contemporary optic routes and options of PMD compensation by using compensators of various types. There is also mentioned the possibility of profiting from the soliton transmission. The last chapter deals with measuring of a real optic route. It includes measured data and its evaluation due to the ITU-T demands.

La matière condensée moderne porte un intérêt particulier pour la classe de matériaux formée par les isolants topologiques. Ils sont différents des isolants typiques par leurs intéressantes propriétés quantiques; ils se comportent comme des isolants dans leur intérieur, mais contiennent des états conducteurs sur leur surface. On peut mieux comprendre le comportement de certains systèmes en matière condensée, tel que les chaînes de polyacétylène, en étudiant un système unidimensionnel simple : le modèle de Su-Schrieffer-Heeger (SSH). Le modèle SSH décrit des fermions sans spin sautant sur un réseau unidimensionnel où les amplitudes de saut alternent d’un site à l’autre. Ce modèle, bien que simpliste, expose les propriétés clés des isolants topologiques tel que les états délocalisés dans tout le réseau ainsi que les états exponentiellement localisés aux frontières du réseau. Dans ce projet, nous étudions le modèle SSH, mais en ajoutant un défaut central dans le réseau qu’on appelle un soliton. Dans notre cas, le soliton consiste en un site central donc les amplitudes de saut sont les mêmes d’un côté et de l’autre. Nous trouvons un ensemble de solutions complet incluant des états de basse énergie localisés aux frontières ainsi que des états de haute énergie localisés au soliton. / Topological insulators are a class of materials that have attracted much attention in modern condensed matter. They are different from typical insulators as they exhibit interesting quantum properties; they behave as insulators in their interior but have conducting states on their surface. We can better understand the properties of low dimensional condensed matter systems (like poly-acetylene chains) by studying a toy model known as the Su-Schrieffer-Heeger (SSH) Model. The SSH model describes spinless fermions hopping on a one-dimensional lattice with staggered hopping amplitudes. Such a toy model exhibits key properties of topological insulators, such as bulk states (delocalized states across the lattice) and edge states (exponentially localized states at the boundaries of the lattice). In this project, we study the SSH model with an added central defect to the chain, which we call a soliton. In our case, the soliton consists of a central site with the same hopping amplitude on either side. We study the impact of such a defect on the properties of the system; we find a complete set of solutions including near-zero-energy edge states as well as high-energy states localized at the soliton.

No description available.

We present Lax pairs and functions called Lax functions corresponding to Calogero- Moser-Sutherland (CMS) systems. We present the Benjamin-Ono (BO) equation and a pole ansatz to the BO equation, constructed from a specific type of Lax function called a special Lax function corresponding to Rational and Trigonometric CMS systems. We present a generalization of the BO equation called the non-chiral Intermediate wave (ncILW) equation and show that a family of solutions to the ncILW equation can be constructed from the special Lax function corresponding to the hyperbolic CMS system. We present the Szegö equation on the circle and the real line. We obtain a family of solutions to the Szegö equation on the real line using a pole ansatz. Using numerical methods, we display solution plots to the BO equation and Szegö equation.

La présente thèse est consacrée à l’étude des peignes optiques de Kerr dans les résonateurs àmodes de galerie, au sein desquels la lumière peut être excitée par pompage externe. L’effet Kerrexistant dans ces résonateurs engendre des modes latéraux équidistants (dans le domaine spectral)de part et d’autre du mode excité, c’est à dire un peigne de fréquence. Cette thèse est diviséeen trois chapitres. Le premier est dédié à l’introduction de la génération de ces peignes et leurapplications. Le deuxième chapitre présente l’analyse de l’équation de Lugiato-Lefever, décrivantde manière analytique le système, et conduit à la construction de deux diagrammes de bifurcationpour les dispersions normale et anomale. Ils sont tracés en fonction des deux seuls paramètresexpérimentalement contrôlables une fois le résonateur fabriqué : la puissance du laser et sondécalage de fréquence. Ces diagrammes indiquent les plages de paramètres pour lesquels une,deux, ou trois solutions existent ainsi que leur stabilité. Les simulations numériques renseignentle type exact de solution associée à chaque aire (notamment les solitons brillants et sombres, lesbreathers, les peignes optiques de Kerr de premier et deuxième ordre, et un régime chaotique) ; cesdiagrammes indiquent donc les paramètres du laser à choisir afin de générer la solution souhaitée.Le troisième chapitre est dédié aux peignes de Kerr optique secondaires, lignes additionnelles dansle domaine spectral générées entre les lignes du peigne principal. Ils apparaissent en dispersionanormale, lorsque la quantité de photon pompe excède un seuil dit de second ordre, qui a étédéterminé numériquement. / This thesis is dedicated to the study of the Kerr optical frequency combs in whispering gallery moderesonators, where the light can be excited by the extern pump. Due to the Kerr effect existing in theseresonators, the quasi-equidistant lines in the spectral domain are generated around the excited mode,that is the frequency comb. This thesis is devided in three chapters. The first one is dedicated to theintroduction of the Kerr comb generation and their applications.The second one presents the analysisof the Lugiato-Lefever equation used for the analytical study of the system, leading to the constructionof two bifurcation diagrams for the normal and anomalous dispersions. They are plotted for twoparameters, which can be controlled during experiments once the resonator has been fabricated,which are the pump power of the laser and its frequency detuning. These diagrams show the areas ofthe parameters for which one, two, or three solutions exist and their stability. The additional numericalsimulations show the exact type of the solution in each area (such as the bright and dark solitons,the breathers, the primary and secondary Kerr combs and chaotical regimes), finally these diagramsshow the parameters of the laser needed to be choosen for the generation of the desired solution.The third chapter is dedicated to the secondary Kerr combs, which are the additional lines generatedbetween the lines of the primary comb. They appear in the anomalous dispersion regime, when thequantity of the pump photons crosses the second-order threshold, which has been found numerically.

Cette thèse porte sur l’étude des propriétés qualitatives des solutions des équations des ondes et de la chaleur semi-linéaires. Les résultats qui y sont décrits sont les suivants. Les deux premiers concernent l’existence et la description de dynamiques explosives de concentration en temps fini de l’état stationnaire à symétrie radiale dans le régime dit énergie surcritique ; en outre, pour l’équation des ondes la stabilité de ces phénomènes est étudiée dans le cas radial, et pour l’équation de la chaleur le cas plus général d’un domaine borné avec conditions de Dirichlet au bord est considéré. Le troisième porte sur la classification des dynamiques possibles près de l’état stationnaire radial pour l’équation de la chaleur dans le régime dit énergie critique, trois scénarios ayant lieu : la stabilisation, l’instabilité par explosion auto-similaire à profil explosif constant en espace, et l’instabilité par dissipation vers la solution nulle. Enfin, le quatrième a pour objet l’existence et la stabilité de profils explosifs auto-similaires non constants en espace pour l’équation de la chaleur dans le cas énergie surcritique / This thesis is devoted to the study of qualitative properties for solutions to the semilinear heat and wave equations. The results that are described are the following. The first two concern the existence and description of blow-up dynamics in which the radially symmetric stationary state is concentrated in finite time in the so-called energy supercritical regime; in addition, for the wave equation the stability of these phenomena is studied in the radial case, and for the heat equation the more general case of a bounded domain with Dirichlet condition at the boundary is considered. The third one deals with the classification of the possible dynamics near the radial stationary state for the heat equation in the so-called energy critical regime, where three scenarii occur: stabilization, instability by blow-up with the constant in space blow-up profile, and instability by dissipation to the null solution. Eventually, in the forth result we investigate the existence and the stability of self-similar blow-up profiles that are not constant in space, for the heat equation in the energy supercritical case