published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy

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The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\"odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.

Cette thèse porte sur la propagation d’ondes non-linéaires dans des métamatériaux acoustiques unidimensionnels. Plus précisément, nous voulons étudier les interactions entre les non-linéarités, les pertes et la dispersion. Ce travail combine des calculs analytiques, des simulations numériques et des résultats expérimentaux. En particulier, nous concentrons notre analyses sur deux phénomènes : la génération du second harmonique et la formation de solitons acoustiques. Deux types différents de métamatériaux sont étudiés : (i) un guide d’onde chargé par une distribution périodique de trous latéraux (milieu à densité effective négative) et (ii) un guide d’onde chargé périodiquement par des  plaques élastiques encastrées (milieu à masse effective négative). En s’appuyant sur une analogie électroacoustique et sur la théorie des lignes de transmission, un modèle discret de la propagation est développé pour chaque système. L’approximation des grandes longueurs d’ondes est ensuite utilisée pour obtenir une modèle continu permettant d’établir une équation non-linéaire, dispersive et dissipative pour la propagation. Cette dernière est analysée à l’aide de la méthode des perturbations conduisant à une expression analytique pour la génération du second harmonique. De plus, la méthode des échelles multiples est utilisée pour obtenir les diverses solutions de solitons d’enveloppe (bright, dark et gray) présents dans les systèmes. Les prédictions analytiques sont corroborées par des simulations numériques directes et des mesures de la génération de second harmonique sont effectuées mettant en lumière un bon accord avec le modèle théorique. / The subject of this PhD thesis is the propagation of nonlinear waves in 1D acoustic metamaterials. More specifically we aim to study the interplay between nonlinearity, loss and dispersion. Our studies combine analytical calculations, numerical simulations and experimental results. In particular we focus our analysis on two main phenomena: the second harmonic generation and the formation of solitary waves. Two different acoustic metamaterials are studied: (i) A waveguide loaded with a periodic distribution of side holes (featuring negative effective bulk modulus) and (ii) a waveguide periodically loaded with clamped elastic plates (featuring negative effective mass density). Relying on the electroacoustic analogy and the transmission line approach, we derive a discrete lattice model for each system. The corresponding long wavelength, continuum approximation of the lattice models, leads to a nonlinear, dispersive and dissipative wave equation. From the latter, by utilising a perturbation method, we obtain analytical results regarding the second harmonic generation. Furthermore with the use of a multiple scale analysis we find various envelope (bright, gap, black and gray) soliton solutions supported by the acoustic metamaterial. The analytical predictions are corroborated by direct numerical simulations. We finally perform experiments on an acoustic waveguide loaded with a periodic distribution of side holes and measure the second harmonic generation in close agreement with our theoretical predictions.

Within the inflationary paradigm, our patch of the universe near the end of inflation is highly homogeneous and isotropic as necessitated by cosmic microwave background observations. This patch, however, is also in a cold and non-thermal state. A successful model of an inflationary primordial universe should account for how the universe transitioned from an inflationary to a radiation-dominated, hot, thermal phase required for the production of light elements via big-bang nucleosynthesis. It is desirable for such a model also to include a mechanism for the generation of the observed matter-antimatter asymmetry and perhaps a primordial mechanism for the generation of cosmic magnetic fields. The transition from an inflationary to a radiation-dominated, thermal phase (reheating) is likely to be phenomenologically rich. Reheating could include explosive particle production and various other non-perturbative, non-linear and non-equilibrium phenomena. Reheating can leave its own observational signatures in the form of gravitational waves and non-Gaussianities. Importantly, reheating can also affect the observational predictions of the preceding phase of inflation. Reheating remains an active field of research, with significant gaps in our understanding of the process. This thesis is an attempt to improve our understanding of the period following inflation, including reheating, through an exploration and analysis of realistic post-inflationary models with the aid of detailed numerical simulations. The focus of the studies is on aspects of the models with potential observational implications. In Part I of this thesis, we provide an overview of inflation and its end, concentrating on our current understanding of reheating and the challenges we face in trying to constrain reheating observationally. In Part II, we consider the post-inflationary expansion history in a broad class of observationally-favoured single-field models of inflation. Generally, the ambiguity in the expansion history of reheating can cause significant uncertainty in predictions for inflationary observables such as the spectral index, n_s, and the tensor-to-scalar ratio, r. The work in this part considers the full non-linear evolution of the inflaton during the initial stages of reheating and places bounds on the post-inflationary expansion history when perturbative couplings of the inflaton to other relativistic fields are included. In Part III, we investigate non-perturbative particle production and non-linear dynamics after inflation in models where the inflaton is charged under global/local symmetries. We first explore the effects of the non-linear inflaton dynamics for the generation of matter-antimatter asymmetry in the case where a global U(1) symmetry of the inflaton is weakly broken. We find a parameter range in which the model successfully predicts the observed baryon-to-photon ratio. We then consider the particle production during and after inflation in models with a charged inflaton under Abelian, U(1), and non-Abelian, SU(2) and U(1) x SU(2), gauge symmetries. Finally, we present a novel algorithm for evolving the full set of coupled, non-linear equations describing the U(1) charged inflaton and accompanying gauge fields on a lattice in an expanding universe. The novel feature here is that the gauge constraints are satisfied to machine precision when the gravitational dynamics are self-consistently included at the background level, and there are no restrictions on the order of the time-integrators.

Neste trabalho estudamos aspecto estruturais de modelos onde os bárions são tratados como sólitons quirais, tais como o proposto pó Skyrme e variantes, contendo um termo estabilizador de sexta ordem, proporcional à corrente bariônica. Modelos deste tipo têm sido bastante estudados na literatura e suas predições para as propriedades estásticas do núcleon, sistematicamente, se mantêm por volta de 2/3 dos respectivos valores experimentais. Em geral, uma dada versão do modelo envolve apenas dois ou três parâmetros, mas pode dar origem a mais de uma dezena de previsões, e esta proliferação de números torna difícil a sua avaliação. Por isso, neste trabalho investigamos os vínculos estruturais ou numéricos entre as previsões do modelo, de modo a tornar mais objetiva a sua comparação com a experimentação. Todos os modelos considerados têm a mesma parte de longo alcance. Assim, a busca de padrões é feita considerando-se tanto versões diferentes das lagrangianas de curto alcance, vários valores para os parâmetros, e campos do píon que se transformam segundo representações não usuais de SU(2) X SU(2). Dessas várias possibilidades emerge um número muito grande de resultados numéricos que, depois de organizados, mostram regularidade. Dessas regularidades decorre uma proposta para a análise dos dados experimentais. Finalmente, este trabalho também inclui um cálculo alternativo do fator de forma píon-núcleon, a partir da interação NN do modelo de Skyrme. / In this work we study structural features of models where baryons are treated as chiral solitons, such as that proposed by Skyrme and variants, containing a sixth order stabilizing term, proportional to the baryonic current. Models of this kind have been widely considered in the literature and their predictions for nucleon static properties are systematically around 2/3 of the corresponding experimental values. In general a given version of the model contains only two or three parameters, but may hield more than ten observable predictions. This makes its assessment rather difficult. Therefore in this work we investigate structural and numerical constraints between the predictions of the model, so as to render the comparision with experimental more objective. All the models considered here contain the same long range parl. Hence the search of patterns is clone by considering different versions for the short range lagrangian. Various values for the free parameters and pion fields which transform according to unusual representations of SU(2) x SU(2) . A rather large amount of numerical results emerge form these various possibilities, which display regularities after being organized. These patterns motivate a proposal for the analysis of experimental information. Finally, this work also includes an altenative calculation for the pion-nucleon form factor, starting from the NN interaction in the Skyrme model.

Orientador: Valery Shchesnavich / Dissertação (mestrado) - Universidade Federal do ABC. Programa de Pós-Graduação em Física, 2012.

Orientador: Marcelo Batista Hott / Coorientador: Alvaro de Souza Dutra / Banca: Denis Dalmazi / Banca: Roberto André Kraenkei / Banca: Othon Cabo Winter / Wesley Bueno Cardoso / Resumo: Nesta tese lidamos com a equação de Schroedinger não linear com coeficientes modulados em diferentes contextos. Esta equação diferencial não linear é amplamente usada para descrever a propagação de pulsos de luz através de uma fibra óptica ou para modelar a dinâmica de um condensado de Bose-Einstein. Primeiro, aplicamos as transformações canônicas de ponto para resolver algumas classes de equação de Schroedinger não linear com coeficientes modulados ou seja, aqueles que possuem não linearidades cúbica e quântica (dependentes do espaço e tempo) específicas. O método aplicado aqui nos permite encontrar soluções tipo sólitons localizados (no espaço) para a equação de Schroedinger não linear com coeficientes modulados, que não foram apresentados antes. No contexto de condensados de Bose-Einstein, nós generalizamos o potencial externo o qual armadilha o sistema, e os termos de não linearidade da equação diferencial. Em seguida, aplicamos as transformações canônicas de ponto para resolver algumas classes de duas equações de Schroedinger não lineares acopladas com coeficientes modula-dos isto é, não linearidades cúbica e quântica - dependentes do espaço e tempo - específicas. O método aplicado aqui nos permite encontrar uma classe de soluções de sólitons tipo vetoriais localizados (no espaço) das duas equações de Schroedinger não linear acopladas. Os sólitons vetoriais encontrados aqui podem ser aplicados a estudos teóricos de condensados de Bose-Einstein de átomos com dois estados internos diferentes ou á propagação de pulsos de luz através de fibras ópticas focalizadoras ou desfocalizadoras. Finalmente, usando transformações canônicas de ponto obtemos soluções exatas localizadas (no espaço) da equação de Schroedinger não linear com não linearidades cúbica e quântica moduladas no espaço e tempo ...(Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this thesis we deal with the nonlinear Schrödinger equation with modulated coefficients in different contexts. This nonlinear differential equation is widely used to describe light pulses propagating through an optical fiber or to model the dynamics of a Bose-Einstein condensate. First, we apply point canonical transformations to solve some classes of nonlinear Schrödinger equation with modulated coefficients namely, those which possess specific cubic and quantic (time- and space-dependent) nonlinearities. The method applied here allows us to find wide localized (in space) soliton solutions to the nonlinear Schrödinger equation, which were not presented before. In the context of Bose-Einstein condensates, we also generalize the external potential which traps the system and the nonlinearities terms. Then, we apply point canonical transformations to solve some classes of two coupled nonlinear Schrödinger equations with modulated coefficients namely, specific cubic and quantic - time and space dependent - nonlinearities. The method applied here allows us to find a class of wide localized (in space) vector soliton solutions of two coupled nonlinear Schrödinger equations. The vector solitons found here can be applied to theoretical studies of Bose-condensed atoms in two different internal states and of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. Finally, we use point canonical transformations to obtain localized (in space) exact solutions of the nonlinear Schrödinger equation with cubic and quantic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) term. We obtain a class of wide localized exact solutions of nonlinear Schrödinger equation in the presence of a number of non-Hermitian ... (Complete abstract click electronic access below) / Doutor

Orientador: Denis Dalmazi / Coorientador: Álvaro de Souza Dutra / Banca: Julio Marny Hoff da Silva / Banca: Rafael Augusto Couceiro Corrêa / Resumo: Certos sistemas físicos podem ser descritos por uma classe de equações não-lineares. Essas equações descrevem pacotes de onda chamado de sólitons que tem aplicações em diversas áreas, por exemplo, Óptica, Cosmologia, Matéria Condensada e Física de Partículas. Alguns métodos foram desenvolvidos ao longo dos anos para encontrar as soluções dessas equações. Buscaremos essas soluções usando o que chamamos de Método Iterativo de Taylor (MIT), que fornece uma solução aproximada em polinômio de Taylor de forma distinta do que se tem na literatura. Usaremos o MIT para calcular soluções por aproximantes de Padé que são razões entre dois polinômios e fornecem soluções melhores que o polinômio de Taylor que o gerou. Inicialmente resolveremos a equação de um modelo de um campo denominado λφ4 . Em seguida resolveremos um modelo com dois campos escalares acoplados e encontraremos uma solução analítica aproximada em casos onde não existe solução analítica, explorando a diversidade das soluções do modelo. Usando essa abordagem por aproximantes de Padé veremos que há algumas vantagens em relação a outros métodos / Abstract: Certain physical systems can be described by a class of non-linear differential equations. Those equations describe wave packets called solitons which have applications in several areas, for example, Optics, Cosmology, Condensed Matter, and Particle Physics. Some methods have been developed over the years to find solutions to these equations. We will look for those solutions using what we call the Taylor Iterative Method (TIM), which provides an approximate solution in terms of a Taylor's polynomial in a unusual way, regarding the present literature. We will use TIM to calculate solutions by Padé approximants, which are ratios between two polynomials and provide better solutions than the Taylor polynomial itself. We first solve the field equation of a model called λφ4 . Then we will solve a model with two coupled scalar fields and find an approximate analytic solution in cases where there is no known analytical solution, exploring the diversity of the solutions of the model. We will see that there are some advantages in using the Padè approximants as compared to other methods / Mestre

In this thesis, the interaction pattern for a class of soliton solutions of the Kadomtsev- Petviashvili (KP) equation (−4ut + uxxx + 6uux )x + 3uyy = 0 is analyzed. The complete asymptotic properties of the soliton solutions for y → ±∞ are determined. The resonance characteristic of two sub-classes of the soliton solutions, in which N- incoming line solitons for y → −∞ interact to form N+ outgoing line solitons for y → ∞, is described. These two specific sub-classes of (N-,N+)-soliton solutions are the following: 1) [(2, 3), (2, 4), (2, 5)], 2) [(3, 2), (3, 3), (3, 4)]. The intermediate solitons and the interaction regions of the above soliton solutions are determined, and their various interaction patterns are explored. Maple and Mathematica are used to get the 3 dimensional plots and contour plots of the soliton solutions to show their interaction patterns. Finally, the spider-web-structures of the discussed solitons of the KP equation are displayed.