Neste trabalho são apresentados alguns resultados matemáticos relevantes para a aplicação do método de espalhamento inverso à resolução de uma classe de equações de evolução não-lineares. É demonstrada a propriedade isoespectral para certas famílias de operadores lineares não auto-adjuntos. Esta propriedade tem um papel central na aplicação do método acima a equações de evolução não-lineares de interesse físico, tais como a equação de sine-Gordon e a equação de Schrödinger não-linear. É feito também, uma teoria de espalhamento inverso rigorosa para sistemas do tipo Zakharov-Shabat, o que inclui uma análise qualitativa do espectro de operadores deste tipo. / This Thesis presents some mathematical results relevant in applications of the inverse scattering transform to the solution of a class of non-linear evolution equations. First, it is shown that certain families of non-selfadjoint linear operators have the isospectral property, which is fundamental for the above applications. These families include various operators related to no-linear equations of great physical interest, such as the sine-Gordon and the non-linear Schrödinger equations. In the sequel, a rigorous inverse scattering theory, including a qualitative spectral analysis, is developed for systems of Zakharov-Shabat type.

Neste trabalho são apresentados alguns resultados matemáticos relevantes para a aplicação do método de espalhamento inverso à resolução de uma classe de equações de evolução não-lineares. É demonstrada a propriedade isoespectral para certas famílias de operadores lineares não auto-adjuntos. Esta propriedade tem um papel central na aplicação do método acima a equações de evolução não-lineares de interesse físico, tais como a equação de sine-Gordon e a equação de Schrödinger não-linear. É feito também, uma teoria de espalhamento inverso rigorosa para sistemas do tipo Zakharov-Shabat, o que inclui uma análise qualitativa do espectro de operadores deste tipo. / This Thesis presents some mathematical results relevant in applications of the inverse scattering transform to the solution of a class of non-linear evolution equations. First, it is shown that certain families of non-selfadjoint linear operators have the isospectral property, which is fundamental for the above applications. These families include various operators related to no-linear equations of great physical interest, such as the sine-Gordon and the non-linear Schrödinger equations. In the sequel, a rigorous inverse scattering theory, including a qualitative spectral analysis, is developed for systems of Zakharov-Shabat type.

Orientador: Patricio Anibal Letelier Sotomayor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T10:00:29Z (GMT). No. of bitstreams: 1 AyalaMolina_JairoAlonso_D.pdf: 1244165 bytes, checksum: 30d706ffeeaefdce6a15f3a200327b18 (MD5) Previous issue date: 2011 / Resumo: Na atualidade, o estudo de objetos como lentes gravitacionais ou buracos negros em dimensões superiores, bem como a formulação de cosmologias de Kaluza-Klein, têm recebido cada vez maior atenção. Na tentativa de compreender melhor estes e outros temas semelhantes, o estudo das soluções exatas, assim como de algoritmos para sua geração, desempenha um papel muito importante. Neste trabalho, apresentamos as equações de Einstein no vácuo para uma classe especial de espaço-tempo D-dimensional que admite D - 2 campos vetoriais de Killing, assim como sua formulação matricial. Apresentamos também a extensão de dois algoritmos apresentados no artigo do professor Patricio Letelier, On the Inverse-Scattering Method Generation of Gravitational Waves and other New Solution-Generating Algorithms, Nuovo Cimento 97 B, 1 (1987), para depois aplicá-los na obtenção de soluções não diagonais. Igualmente estudamos a aplicação do método de Belinski-Zakharov para a obtenção de soluções solitônicas multidimensionais a partir de nossa métrica, que admite representação diagonal por blocos. Finalmente, aplicamos os algoritmos de geração apresentados às métricas de Kaluza-Klein para obter novas soluções das correspondentes teorias efetivas em quatro dimensões, assim como seus tensores de energia-momento. Exemplos de possíveis interpretações destes tensores na teoria clássica de campos (ClFT) e na mecânica de fluidos, são apresentados também / Abstract: Nowadays, the study of objects such as black holes or gravitational lenses in higher dimensions, as well as the formulation of Kaluza-Klein cosmologies, have received increasing attention. In an attempt to better understand these and other similar topics, the study of exact solutions and the algorithms for their generation, plays a very important role. In the present work we present the Einstein equations in vacuum for a special class of D-dimensional space-time which admits D - 2 Killing vector fields, as well as its matrix formulation. We also present the extension of two algorithms studied in the Patricio Letelier's paper On the Inverse-Scattering Method Generation of Gravitational Waves and other New Solution-Generating Algorithms, Nuovo Cimento 97 B, 1 (1987), to later apply them in obtaining non-diagonal solutions. We also studied the method of Belinski-Zakharov to obtain multi-dimensional soliton solutions from our metric, which admits representation diagonal by blocks. Finally, we apply the presented generation algorithms to Kaluza-Klein metrics to obtain new solutions of the corresponding effective theories in four dimensions, as well as its energy-momentum tensors. Examples of possible interpretations of these tensors in classical field theory (ClFT) and fluid mechanics, are also presented / Doutorado / Fisica-Matematica / Doutor em Matemática Aplicada

Orientadores: Patricio A. Letelier Sotomayor e Marcelo Moraes Guzzo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-13T19:55:09Z (GMT). No. of bitstreams: 1 Castro_GianMachadode_D.pdf: 3217878 bytes, checksum: 48c026fc06d4c9e5db03014506ffc609 (MD5) Previous issue date: 2009 / Resumo: Nesta tese, estudamos o problema de um anel delgado de matéria de densidade constante com um buraco negro de Kerr em seu centro. Nosso objetivo foi resolver as equações de Einstein no vácuo com simetria axial para esse sistema gravitacional. Para fazer a sobreposição não-linear do anel com o buraco negro (BN), utilizamos o método de Belinsky e Zakharov (MBZ). Este método necessita de uma solução conhecida (solução semente) para gerar uma nova solução. Tomamos a aproximação da solução do anel em multipolos como solução semente. Como resultado, obtivemos a solução de um anel com o BN central. A expansão do anel em multipolos exige o truncamento da série. Esta aproximação introduz um erro em nossa solução. Realizamos o estudo do mesmo devido ao truncamento da série. Também estudamos a estabilidade de órbitas circulares equatoriais de partículas movendo-se ao redor do sistema anel-BN quanto a perturbações epicíclicas e verticais. Analisamos essas perturbações para os modelos de gravitação relativística e newtoniana. Como resultado, encon- tramos o efeito inesperado da duplicação das órbitas circulares de flotons para alguns valores de parâmetros relacionados com o anel e o BN, bem como zonas de estabilidade na região interna do anel. / Abstract: In this thesis, we will study the problem of a thin ring of matter of constant density with a central Kerr black hole. The aim of this work is to solve the Einstein equations in the vacuum with axial symmetry for that gravitational system. To do the nonlinear superposition of the ring with the black hole (BH), we used the Belinsky and Zakharov method (BZM). This method needs a known solution (called seed solution) to generate a new one. We took the Newtonian ring potential approximated by a multipolar expansion as seed solution. As result, we obtained the solution of a ring with a central BH. The ring multipolar expansion demands the truncation of the series. This approach introduces an error in our solution. Estimations of errors due to the truncation of the multipolar expansions are performed. We also studied the stability of equatorial circular orbits of particles moving around the system ring plus BH due to epicycle and vertical perturbations. We analyzed those perturbations for relativistic and Newtonian gravitational models. As result, we found the unexpected effect of the duplication of the photons circular orbits for certain values of parameters related with the ring and BH, as well as zones of stability in the inner area of the matter ring. / Doutorado / Relatividade e Gravitação / Doutor em Ciências

Inspiriert von den integrablen Strukturen der schwach gekoppelten planaren N=4 Super-Yang-Mills-Theorie studieren wir Q-Operatoren und Yangsche Invarianten. Wir geben eine Übersicht der Quanten-Inverse-Streumethode zusammen mit der Yang-Baxter Gleichung welche zentral für diesen systematischen Zugang zu integrablen Modellen ist. Den Fokus richten wir auf rationale integrable Spinketten und Vertexmodelle. Wir besprechen einige ihrer bekannten Gemeinsamkeiten und wie sie durch Bethe-Ansatz-Methoden mit Hilfe sogenannter Q-Funktionen gelöst werden können. Der Hauptteil basiert auf den ursprünglichen Publikationen des Autors. Zuerst konstruieren wir Q-Operatoren, deren Eigenwerte zu den Q-Funktionen rationaler homogener Spinketten führen. Die Q-Operatoren werden als Spuren gewisser Monodromien von R-Operatoren eingeführt. Unsere Konstruktion erlaubt es uns die Hierarchie der kommutierenden Q-Operatoren und ihre funktionalen Beziehungen herzuleiten. Wir studieren wie der nächste-Nachbarn Hamiltonoperator, sowie höhere lokale Ladungen direkt aus den Q-Operatoren extrahiert werden können. Danach widmen wir uns der Formulierung der Yangschen Invarianzbedingung, wie sie auch im Zusammenhang mit Baumgraphen die bei der Berechnung von Streuamplituden in der N=4 Super-Yang-Mills-Theorie auftreten, innerhalb der RTT-Realisierung. Dies erlaubt es uns den algebraischen Bethe-Ansatz anzuwenden und die dazugehörigen Bethe Gleichungen herzuleiten, welche für die Konstruktion der Eigenzustände die Yangsche Invarianz aufweisen, relevant sind. Die Komponenten dieser Eigenzustände der von uns betrachteten Spinketten können außerdem als Zustandssummen gewisser zweidimensionaler Vertexmodelle angesehen werden. Zudem analysieren wir die Verbindung zwischen den Eigenzuständen und den oben genannten Baumgraphen. Schlussendlich diskutieren wir die von uns vorgelegten Ergebnisse und deren Folgen im Hinblick auf die Erforschung der planaren N=4 Super-Yang-Mills-Theorie. / Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method QISM along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author''s original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain two-dimensional lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions and implications of our studies for planar N=4 super Yang-Mills theory.

Diese Arbeit untersucht die N = 4 super-Yang-Mills-Theorie bei schwacher Kopplung, mit dem Ziel eines tieferen Verständnisses von Größen der Theorie als Zustände des integrablen Modells dass der planaren Theorie zu Grunde liegt. Wir leiten On-Shell-Diagramme für Formfaktoren des chiralen Energie-Impuls-Tensor-Multipletts aus der BCFW-Rekursion her, und untersuchen deren Eigenschaften. Dies erlaubt die Herleitung eines Graßmannschen Integrals. Für NMHV-Formfaktoren bestimmen wir die Integrationskontur. Dies erlaubt es das Integral mit einer Twistor-String-Formulierung in Beziehung zu setzen. Mit Hilfe dieser Methoden zeigen wir dass Formfaktoren des chiralen Energie-Impuls-Tensor-Multipletts und On-Shell-Funktionen mit Einfügungen beliebiger Operatoren Eigenzustände integrabler Transfermatrizen sind. Diese Identitäten verallgemeinern die Yangsche Invarianz der On-Shell-Funktionen von Amplituden. Wir zeigen weiterhin dass ein Teil der Yangschen Symmetrien erhalten bleibt. Wir erweitern unsere Untersuchung auf nichtplanare On-Shell-Funktionen und zeigen dass sie ebenfalls solche Symmetrien besitzen. Weitere Identitäten mit Transfermatrizen werden hergeleitet, und zeigen insbesondere dass Diagramme auf Zylindern als Intertwiner fungieren. Als Schritt hin zur Berechnung der Eigenzustände des integrablen Modells zu höheren Schleifenordnungen untersuchen wir Einspuroperatoren. Hier erlaubt die Quantum Spectral Curve die nichtperturbative Berechnung ihres Spektrums, liefert jedoch keine Information zu den Zustände. Die QSC kann als Q-System verstanden werden, welches durch Baxter Q-Operatoren formulierbar sein sollte. Um darauf hinzuarbeiten untersuchen wir die Q-Operatoren nichtkompakter Superspinketten und entwickeln ein effiziente Methode zur Berechnung ihrer Matrixelemente. Dies erlaubt es das gesamte Q-System durch Matrizen für jeden Anregungssektor zu realisieren, und liefert die Grundlage für perturbative Rechnungungen mit der QSC in Operatorform. / This thesis investigates weakly coupled N = 4 super Yang-Mills theory, aiming at a better understanding of various quantities as states of the integrable model underlying the planar theory. We use the BCFW recursion relations to develop on-shell diagrams for form factors of the chiral stress-tensor multiplet, and investigate their properties. The diagrams allow to derive a Graßmannian integral for these form factors. We devise the contour of this integral for NMHV form factors, and use this knowledge to relate the integral to a twistor string formulation. Based on these methods, we show that both form factors of the chiral stress-tensor multiplet as well as on-shell functions with insertions of arbitrary operators are eigenstates of integrable transfer matrices. These identities can be seen as symmetries generalizing the Yangian invariance of amplitude on-shell functions. In addition, a part of these Yangian symmetries are unbroken. We furthermore consider nonplanar on-shell functions and prove that they exhibit a partial Yangian invariance. We also derive identities with transfer matrices, and show that on-shell diagrams on cylinders can be understood as intertwiners. To make progress towards the calculation of the higher loop eigenstates of the integrable model, we consider single trace operators, for which the Quantum Spectral Curve determines their spectrum non-perturbatively. This formulation however carries no information about the states. The QSC is an algebraic Q-system, for which an operatorial form in terms of Baxter Q-operators should exist. To initiate the development such a formulation we investigate the Q-operators of non-compact super spin chains and devise efficient methods to evaluate their matrix elements. This allows to obtain the entire Q-system in terms of matrices for each magnon sector. These can be used as input data for perturbative calculations using the QSC in operatorial form.