Orientador: Luiz Agostinho Ferreira / Banca: Abraham Hirsz Zimerman / Banca: Daniel Augusto Turolla Vanzella / Banca: Francisco Castilho Alcaraz / Banca: Antonio Lima Santos / Resumo:Nosso objeto de estudo são teorias clássicas de campo que possuem sólitons topológicos e um número infinito de quantidades conservadas. Em particular, nossos modelos exibem a assim chamada "carga de Hopf". Esta carga surge porque, para soluções com energia finita, nossas teorias definem mapeamentos de um espaço-tempo compactificado em um espaço alvo 'S POT. 2'. Demonstramos que o conjunto de quantidades conservadas de nossos modelos está relacionado com a invariância da Lagrangiana sob os difeomorfismos de área do espaço alvo. Nossos modelos são bastante relacionados com o modelo de Skyrme-Faddeev e, portanto, apresentamos uma breve introdução a este modelo. Usando o método de Lie, foram descobertas as simetrias das equações de Euler-Lagrange para os citados modelos, para um espaço-tempo curvado genérico. A condição de simetria se relaciona com a solução da equação de Killing para um dado espaço-tempo. Então as equações correspondentes são solucionadas para alguns exemplos específicos como os espaços-tempos euclidiano, 'S POT. 3' X R e de Minkowski. A seguir, para os modelos em 'S POT. 3' X R, usando as simetrias recém descobertas, encontramos sistemas de coordenadas para os quais existem ansatze levando à separação de variáveis e a redução das equações (inicialmente EDPs) à EDOs. As EDOs são lineares e, portanto, sua resolução é obtida a partir da qual calculamos todas as quantidades físicas relevantes como a energia, a carga de Hopf e as cargas de Noether. Também explicamos porque o ansatz escolhido leva à uma EDO linear em nosso caso e porque o algoritmo de Lie funciona / Abstract: ur object of study are classical field theories which possesses topological solitons and have a infnite number of conserved quantities .In particular our models have what is know as "Hopf charge". This charge appears because, for finite energy solutions, our theories define mappings of a compactified space time in a 'S POT. 2' target space. We show that ours models's set of conserved quantities are related to the invariance of the Lagrangean under area preserving diffeomorphisms of the target space. Our models are closely related to the Skyrme-Faddeev model and so we give a brief introduction to it. Using Lie's method we find the symmetries of the Euler-Lagrange equations of such models, for an arbitrary curved space time. The symmetry condition turns out to be related with the solution of the Killing equations in a given space time. We then solve the corresponding equations for some specific examples, like the Euclidean, Minkowski and the 'S POT. 3' X R space times. Then, for the'S POT. 3' X R models, using the symmetries already found, we are able to find systems of coordinates for which then exists ansätze leading to separations of variables and to the reduction of the Euler-Lagrange equations (initially PDEs) to ODEs. These ODEs are linear and so we are able to integrate then and also to calculate all of the physically meaningful conserved quantities, as the energy, Hopf charge, angular momentum. We also explain why such ansatz leads to a linear ODE in this particular case and why the Lie integration algorithm works / Doutor

Supondo que o núcleo possa ser tratado como um fluido perfeito, nós estudamos as condições para a formação de solitons de Korteweg-de Vries (KdV) na matéria nuclear. A existência de solitons de KdV depende da equação de estado nuclear que, por sua vez, depende da teoria microcóspica subjacente da interação núcleon-núcleon e das aproximações feitas durante os cálculos. No nosso trabalho, nós retomamos estudos sobre solitons no núcleo feitos no passado e substituímos a equação de estado usada anteriormente por outra mais moderna e mais realista, baseada no modelo de Walecka e suas variantes. Nossa análise mostra que solitons de KdV podem ser formados no interior do núcleo com largura em torno de um a dois fermis. / Assuming that the nucleus can be treated as a perfect fluid we study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. The existence of these solitons depends on the nuclear equation of state, which, in its turn, depends on the underlying microscopic theory of the nucleon-nucleon interaction and also on the approximations used in the calculations. In this work we reexamine early works on nuclear solitons, replacing the old equations of state by others, more modern and more realistic, base on QHD and on its variants. Our analysis shows that KdV solitons may indeed be formed in the nucleus with a width around one and two fermis.

Dans cette thèse, nous étudions l'équation de Landau-Lifshitz avec une anisotropie planaire en dimension un. Cette équation décrit la dynamique de l'aimantation dans des matériaux ferromagnétiques. Elle admet des solutions particulières de type onde progressive appelées solitons.D'abord, nous montrons la stabilité asymptotique des solitons de vitesse non nulle appelés solitons sombres dans l'espace d'énergie. Plus précisément, nous prouvons que toute solution correspondant à une donnée initiale proche du soliton de vitesse non nulle, converge faiblement dans l'espace d'énergie en temps long, vers un soliton de vitesse non nulle, sous les invariances géométriques de l'équation. Notre analyse repose sur les idées développées par Martel et Merle pour les équations de Korteweg-de Vries généralisées. Nous utilisons la transformée de Madelung pour étudier le problème dans le cadre hydrodynamique. Nous invoquons ensuite la stabilité orbitale des solitons et la continuité faible du flot afin de construire le profil limite. Nous établissons de plus une formule de monotonie pour le moment, ce qui nous permet d'avoir la localisation du profil limite. Sa régularité et sa décroissance exponentielle découlent d'un résultat de régularité pour les solutions localisées des équations de Schrödinger. Nous finissons la preuve par un théorème de type Liouville, qui nous indique que seuls les solitons vérifient ces propriétés dans leurs voisinages.Nous nous intéressons également à la stabilité asymptotique d'une superposition de plusieurs solitons appelées multi-solitons. Les solitons de vitesse non nulle sont ordonnés selon leurs vitesses et sont initialement bien séparés. Nous démontrons la stabilité asymptotique autour et entre les solitons. Plus précisément, nous montrons que pour une donnée initiale proche de la somme de $N$ solitons sombres, la solution correspondante converge faiblement vers un des solitons de la somme, quand elle est translatée au niveau du centre de ce soliton, et converge faiblement vers zéro quand elle est translatée entre les solitons. / In this thesis, we study the one-dimensional Landau-Lifshitz equation with an easy-plane aniso-tropy. This equation describes the dynamics of the magnetization in a ferromagnetic material. It owns travelling-wave solutions called solitons.We begin by proving the asymptotic stability in the energy space of non-zero speed solitons More precisely, we show that any solution corresponding to an initial datum close to a soliton with non-zero speed, is weakly convergent in the energy space as time goes to infinity, to a soliton with a possible different non-zero speed, up to the geometric invariances of the equation. Our analysis relies on the ideas developed by Martel and Merle for the generalized Korteweg-de Vries equations. We use the Madelung transform to study the problem in the hydrodynamical framework. In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile. We next derive a monotonicity formula for the momentum, which gives the localization of the limit profile. Its smoothness and exponential decay then follow from a smoothing result for the localized solutions of the Schrödinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.We also establish the asymptotic stability of multi-solitons. The solitons have non-zero speed, are ordered according to their speeds and have sufficiently separated initial positions. We provide the asymptotic stability around solitons and between solitons. More precisely, we show that for an initial datum close to a sum of $N$ dark solitons, the corresponding solution converges weakly to one of the solitons in the sum, when it is translated to the centre of this soliton, and converges weakly to zero when it is translated between solitons.

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Approved for public release; distribution is unlimited / Internal solitary waves, or solitons, are often generated in coastal or continental shelf regions when tidal currents advect stratified water over bathymetric relief, creating an internal tide which non-linearly evolves into one or more solitons. A major consequence of solitons in a stratified environment is the vertical displacement of water parcels which can lead to sound speed variability of order 10m/s with spatial scales of order 100 meters and timescales of order minutes. Thus significant variations in sonar performance on both surface based ships and submarines can be expected. An understanding into the nature of acoustic propagation through these waves is vital for future development of sonar prediction systems. This research investigates acoustic normal mode propagation through solitons using a 2D parabolic equation simulation and weak acoustic scattering theory whose primary physics is a single scatter Bragg mechanism. To simplify the theory, a Gaussian soliton model is developed that compares favorably to the results from a traditional sech2 soliton model. The theory of sound through a Gaussian soliton was then tested against the numerical simulation under conditions of various acoustic frequency, source depths, soliton position relative to the source and soliton number. The theoretical results compare favorably with numerical simulations at 75, 150 and 300-Hz. Higher frequencies need to be tested to determine the limits of the first order theory. Higher order theory will then be needed to address even higher frequencies and to deal with weakly excited modes. This research is the first step in moving from a state of observing acoustic propagation through solitons, to one of predicting it. / Outstanding Thesis / Royal Australian Navy author

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