We study the nonlinear Schrödinger (NLS) equation on star graphs with the Neumann- Kirchhoff (NK) boundary conditions at the vertex. We analyze the stability of standing wave solutions of the NLS equation by using different techniques. We consider a half-soliton state of the NLS equation, and by using normal forms, we prove it is nonlinearly unstable due to small perturbations that grow slowly in time. Moreover, under certain constraints on parameters of the generalized NK conditions, we show the existence of a family of shifted states, which are parametrized by a translational parameter. We obtain the spectral stability/instability result for shifted states by using the Sturm theory for counting the Morse indices of the shifted states. For the spectrally stable shifted states, we show that the momentum of the NLS equation is not conserved which results in the irreversible drift of the family of shifted states towards the vertex of the star graph. As a result, the spectrally stable shifted states are nonlinearly unstable. We also study the NLS equation on star graphs with a delta-interaction at the vertex. The presence of the interaction modifies the NK boundary conditions by adding an extra parameter. Depending on the value of the parameter, the NLS equation admits symmetric and asymmetric standing waves with either monotonic or non-monotonic structure on each edge. By using the Sturm theory approach, we prove the orbital instability of the standing waves. / Thesis / Doctor of Philosophy (PhD)

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La distribution des ions et des charges aux interfaces solide-liquide est d'une importance capitale dans de nombreux phénomènes et processus. En particulier, le développement de la microfluidique a renforcé la nécessité de comprendre et de contrôler les distributions d'ions dans la double couche électrique. En effet, les phénomènes qui se produisent dans la couche diffuse de Gouy-Chapman deviennent dominants dans la détermination des propriétés d'écoulement lorsque les tailles de canaux deviennent comparables à la longueur de Debye. Dans ce travail de thèse, nous avons utilisé la technique des ondes stationnaire de rayons X pour déterminer la distribution interfaciale d’ions. Nous avons développé des cellules de quelques centaines de nanomètres voir moins dont l’épaisseur est mesurée précisément par réflectivité des rayons X. Dans ces cellules, avons confiné le film liquide entre les multicouches qui permettent de créer l'onde stationnaire et un film polymère. La fluorescence des ions dans ce champ permet alors d’extraire leur distribution aux interfaces avec une résolution spatiale de quelques Angströms. Ce travail a été réalisé pour plusieurs solutions contenant différents mélanges d'ions monovalents (Chlorure, Potassium, Césium et Iode) à plusieurs concentrations. Dans une deuxième partie de ce travail, nous présentons les résultats pour des chlorures de lanthanides. Ces études nous ont permis de déterminer la composition de la couche de Stern et la distribution des ions au sein de cette couche pour l'ensemble des paires La3+/Ce3+, La3+/Eu3+, La3+/Gd3+, Ce3+/Eu3+, Ce3+/Gd3+ et Eu3+/Gd3+. Nous avons pu alors classer ces différents ions et discuter le classement. / The distribution of ions and charges at solid-water interfaces is of key importance in a number of phenomena and processes. In particular the development of microfluidics has strengthened the need for understanding and controlling ion distributions in the electric double layer. Indeed, phenomena occurring within the diffuse Gouy-Chapman layer become dominant in determining flow properties as channel size become on the same order of magnitude of the Debye length. In this thesis, we used the technique of X-ray standing waves to determine the distribution of interfacial ions. We have developed cells with thicknesses of a few hundred nanometers or less, precisely measured by X-ray reflectivity. The liquid film is confined between the multilayer which create the standing wave and a polymer film. The ion fluorescence allows to determine the interfacial distribution with a spatial resolution of a few Angstroms. This work was carried out for solutions containing different mixtures of monovalent ions (chloride, potassium, cesium and iodine) at several concentrations.In the second part of this work, we present the results for lanthanide chlorides. These studies have allowed us to determine the composition of the Stern layer and the distribution of ions in this layer for all pairs La3+/Ce3+, La3+/Eu3+, La3+/Gd3+, Ce3+/Eu3+, Ce3+/Gd3+ et Eu3+/Gd3+. We could then classify these ions and discuss the series.

The purpose with this project is to give insight about the theory behind, and the usage of, membrane absorption. This was done in an effort to reduce a particular acoustical problem regarding a room mode which resonated strongly in my music studio and was causing trouble within the low frequencies. In this essay I go through the history, theory, making-of and results of said absorption technique, and explore the possibilities of it working as a relatively cheap and space efficient solution to problems regarding standing waves. The results were positive considering the low budget. The standing wave decay time was almost halved and the frequency response improved, which proves the capability of this type of absorption.

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Made available in DSpace on 2015-05-15T11:46:25Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 552096 bytes, checksum: 3c146d673e17bf5ffda76282ad07c24d (MD5) Previous issue date: 2010-08-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study existence of both one-sign and nodal positive solutions (with exactly two nodal domains) for a class of quasilinear Schrödinger equations, which model physic phenomena, for example, in plasma physics. To obtain the results, it was used, mainly, the Nehari method, as well as, regularity theory of elliptic and Concentration-Compactness Principle. / Nesta dissertação, estudamos a existência de soluções positivas e mudando de sinal (tendo exatamente dois domínios nodais) para uma classe de equações de Schrödinger quase lineares, as quais modelam fenômenos físicos, por exemplo, em Física dos Plasmas. Na obtenção dos resultados, foi usado, principalmente, o método de Nehari, bem como teoria de regularidade elíptica e o Princípio de Concentração-Compacidade de P. L. Lions.

Orientadores: Sandra Lucia da Cruz, João A. F. R. Pereira / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Quimica / Made available in DSpace on 2018-07-31T22:23:21Z (GMT). No. of bitstreams: 1 Souza_ArlanLucasde_D.pdf: 3739716 bytes, checksum: 40314e4f8359d3c06b6df822abda9f09 (MD5) Previous issue date: 2002 / Resumo: Uma nova técnica de detecção de vazamentos em tubulação é apresenta­da. A técnica é baseada na análise espectral de sinais de pressão medidos em trechos da tubulação que favorecem a formação de ondas estacionárias. Sendo baseada no espectro do sinal, e não na sua representação temporal, a análise possibilita a identificação do vazamento mesmo quando este ocorre durante a realização de operações que naturalmente encobririam o transiente de pressão gerado pelo vazamento. Testes experimentais foram realizados numa tubulação de 1250 m de comprimento em várias condições de fluxo e configurações de vazamento durante a partida e a parada de uma bomba centrífuga. Transientes de pressão foram obtidos por quatro transdutores conectados a um computador padrão PC comandado por um software de aquisição de dados e controle desenvolvido. Os testes mostraram que a análise espectral dos transientes de pressão, juntamente com o conhecimento dos pontos de reflexão na tubulação, consiste em um método simples e eficiente de se detectar vazamentos / Abstract: The development and test of a technique for leak detection in pipelines is presented. The technique is based on the spectral analysis of pressure signals measured in pipeline sections where the formation of stationary waves is favoured, allowing leakage detection during the start stop of pumps. Experimental tests were performed in a 1250 m long pipeline for various operational conditions of the pipeline (liquid flow rate and leakage configuration). Pressure transients were obtained by four transducers connected to a PC compute. The obtained results show that the spectral analysis of pressure transients, together with the knowledge of reflection points provide a simple and efficient way of identifying leaks during the start stop of pumps in pipelines / Doutorado / Sistemas de Processos Quimicos e Informatica / Doutor em Engenharia Química

This master’s thesis explores the possibility of calculating and using the pressure andvelocity fields responsible for acoustophoresis. The goal was to use simulated data,similar to two-dimensional particle tracks from a specific microfluidic platform, toestimate the force potential and solve the partial differential equation (PDE) thatgoverns the relationship between the force potential and the acoustic fields. Lastly,the possibility of identifying the mechanical properties of unknown particles, once theacoustic field was known, was investigated. The thesis found that solving the PDE usingthe finite difference method was likely not possible and an alternative method has beensuggested. It also found that the use of particle tracks to measure the compressibilityand density of cells as a biomarker is promising, as most simulated particles wereaccurately measured. / Denna uppsats utforskar möjligheten att bestämma och använda de tryck- och hastighetsfält som är ansvariga för akustofores. Målet var att använda simulerad data, likt två-dimensionella partikelbanor för en specifik mikrofluidisk plattform, för att uppskatta kraftpotentialen och lösa den partiella differentialekvation (PDE) som styr relationen mellan kraften och de akustiska fälten. Möjligheten att bestämma mekaniska egenskaper hos okända partiklar, när det akustiska fältet är känt, utreddes också. Slutsatsen är att det inte var möjligt att lösa PDEn med hjälp av finita differensmetoden och istället föreslås en alternativ metod. Preliminära simulerade tester att karaktärisera okända partiklar gav positiva resultat, givet att de akustiska fälten är kända i två dimensioner.

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CAPES / Comuns em projetos de engenharia elétrica, bobinas solenoidais com núcleo de ar e camada simples de enrolamento podem apresentar frequência de autorressonância por efeitos de onda estacionária. A compreensão e determinação destas frequências é portanto de primordial interesse para os profissionais envolvidos com estes componentes. Neste trabalho é apresentada uma teoria que visa explicar o comportamento deste componentes quando em autorressonância por onda estacionária, sendo também desenvolvida uma equação prática que pode determinar tais valores de frequência. O comportamento da impedância nestes componentes também é discutido e demonstrado, onde, baseado nesta teoria, duas novas configurações são apresentadas: uma bobina com ambos terminais abertos e uma bobina com ambos terminais aterrados. Através de uma abordagem didática, correlaciona-se o enrolamento com um tubo de ar ressonante, e explica-se o que é registrado em termos de impedância e fase em um analisador vetorial de impedâncias ao conectar em seus terminais um enrolamento que apresenta autorressonância por onda estacionária. Uma análise da transferência de energia sem fio em circuitos ressonantes magneticamente acoplados é também apresentada, onde é destacada a relação entre eficiência, transferência relativa de potência e impedância refletida em circuitos com 2 e 4 bobinas. Um sistema de transferência de energia sem fio, utilizando bobinas bifilares abertas, é proposto. O sistema possui algumas vantagens uma vez que, a bobina bifilar aberta, ao contrário de uma bobina convencional, possui uma autorressonância série. Portanto, o sistema se comporta como um circuito contendo indutor e capacitor em série. Ainda, uma vez que o circuito dispensa o uso de capacitores, possibilita-se uma menor dimensão das unidades transmissora e receptora, além de se evitar eventuais danos aos capacitores em aplicações envolvendo maiores potências. / Common in electrical engineering projects, air-core single-layer solenoid coils can present self-resonant frequencies due to stationary waves effects. The comprehension and determination of these frequencies is of primordial interest for the professionals evolved with these components. In this work, a theory that aims to explain these components behavior at standing wave self-resonance is presented, a practical equation that can determine such frequency values is also developed. The impedance behavior in these components is also discussed and demonstrated, where, based in this theory, two new configurations are presented: a coil with both terminals open and a coil with both terminals grounded. Through a didactic approach the coil and a resonant air-pipe are correlated, and it is explained what is registered in terms of impedance and phase on a vector impedance analyzer when a standing wave self-resonant coil is connected on its terminals. An analysis of the wireless energy transfer in magnetically coupled resonant circuits is also presented, being emphasized the relation between efficiency, relative power transfer and reflected impedance in 2 and 4-coils circuits. A wireless power transfer system using open bifilar coils is proposed. The proposed system has the advantages that the open bifilar coil, different from a conventional coil, has a series self-resonance. Thus, the system behaves as a series-inductor capacitor circuit. Also, since the circuit avoids capacitors, it allows a smaller transmitter and receptor units dimensions, besides avoiding eventual damages to capacitors in higher power applications.