Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Crossing symmetry relates scattering and annihilation processes to each other. Its derivation is straightforward in perturbative approaches to quantum field theory: it merely reflects the exchange of in- and outgoing states in Feynman diagram computations. In soliton models, the situation is much more complicated because the scattering and the annihilation processes concern distinct topological sectors that are not related by any continuous transformation. In this thesis a simple soliton model will be employed to address this problem numerically. First, in the unit topological sector we extract asymptotically the phase shift of the scattering process of a wave packet off the kink-solution. To this end we solve the time-dependent equation of motion of the non-integrable '4 field model in (1+1) spacetime dimensions for two distinct initial conditions: the wave packet in a trivial vacuum background and in the background of the kink-solution. Second, in the topologically trivial sector we present numerical solutions of the kink– antikink interaction in the same model. We find that the final state of this interaction varies dramatically with the impact velocity. As result, we analyze our numerical solutions for the kink–antikink collisions system in two regimes. For the initial velocity of the system less than some critical velocity, vc 0:26, the kink and the antikink either annihilate or inelastically scatter. On the other hand, the kink and the antikink always inelastically scatter when the initial velocity of the system is higher than this critical velocity. However, the scattering processes of the kink–antikink with initial velocity below and above the critical velocity are different. Below the critical velocity the kink and the antikink collide and always undergo n-bounces (n 2) before they depart to infinity. When the initial velocity of the system is higher than vc, the kink and the antikink depart to infinity after only one bounce. We present a qualitative description for these bounce effects between the kink and the antikink motivated by earlier studies as well as our numerical simulations. We utilize collective coordinates to study the dynamics of the kink–antikink system in two degrees of freedom. In this regime, we modify the ansätze of the kink–antikink system from earlier studies to account for relativistic effects. We perform a comparison between this approximation and the full system. We end our discussion of this sector by discussing the scattering data for the inelastic scattering and the annihilation processes of the kink–antikink. Third, we compare the extracted scattering data for the scattering process of a wave packet off the kink-solution and the annihilation process of the kink–antikink to each other. Finally, these studies of different sectors allow us to make a conjecture about the validity of crossing symmetry within the non-integrable '4 field model. / AFRIKAANSE OPSOMMING: Kruising-simmetrie beskryf ’n verband tussen verstrooiings- en vernietigingsprosesse. Die afleiding daarvan binne die raamwerk van steuringsteorie is eenvoudig: dit behels bloot die omruil van ingaande en uitgaande toestande in die Feynman-diagram. In soliton-modelle is die situasie egter meer ingewikkeld aangesien die verstrooiings- en vernietigingsprosesse in verskillende topologiese sektore plaasvind wat nie deur kontinue transformasies aan mekaar gekoppel is nie. In hierdie tesis word daar van ’n eenvoudige soliton-model gebruik gemaak om hierdie probleem numeries te ondersoek. Eerstens word die faseverskuiwing van die verstrooiingsproses van ’n golfpakkie vanaf ’n kinkoplossing asimptoties in die topologiese eenheidssektor bepaal. Vir hierdie doel word die tydafhanklike bewegingsvergelykings van die klassieke, nie-integreerbare 4-veldeteorie in (1+1) dimensionele ruimte-tyd opgelos. Twee beginkondisies word ondersoek: ’n golfpakkie in die triviale vakuum agtergrond asook in die kinkoplossing agtergrond. Tweedens ondersoek ons ook numeriese oplossings vir die kink-antikink wisselwerking binne die triviale topologiese sektor van dieselfde model. Hier vind ons dat die finale toestand van hierdie wisselwerkingsproses op ’n uiters sensitiewe wyse van die impaksnelheid afhang. Ons ondersoek gevolglik die numeriese oplossings vir die kink-antikink botsings in twee gebiede. Vir beginsnelhede onder die kritieke snelheid vc 0:26 sal die kink en antikink mekaar óf vernietig óf nie-elasties verstrooi. In teenstelling hiermee sal die kink-antikink altyd nie-elastiese verstrooiing ondergaan as die beginsnelheid die kritieke snelheid oorskry. Die aard van die verstrooiingsprosesse vir beginsnelhede bo en onder die kritieke snelheid is egter verskillend. Onder die kritieke snelheid sal die kink en antikink ’n n-bots proses (n 2) ondergaan voor hulle finaal van mekaar weg beweeg. Bo die kritieke snelheid sal die kink-antikink egter net ’n enkele botsing ondergaan en dan uitmekaar beweeg. Ons lewer ’n kwalitatiewe beskrywing vir die bons-effek tussen die kink en antikink wat deur vorige studies asook ons numeriese resultate gemotiveer word. Ons maak gebruik van ’n kollektiewe koördinaatstelsel om die dinamika van die kink-antikink in terme van twee vryheidsgrade te bestudeer. In hierdie gebied pas ons ook die ansatz vir die kink-antikink stelsel aan om relatiwistiese effekte in ag te neem. Ons vergelyk dan hierdie benadering met die oplossing van die volle sisteem. Die bespreking van hierdie sektor word afgesluit met ’n analise van die verstrooiingsdata vir die verstrooiing- en vernietingsprosesse van die kink-antikink. Derdens vergelyk ons die verstrooiingsdata vir die verstrooiing van ’n golfpakkie vanaf ’n kinkoplossing met die van die vernietigingsproses van die kink-antikink. Ons studie van die verskillende sektore laat ons dan toe om ’n vermoede te formuleer oor die geldigheid van kruissing-simmetrie binne die nie-integreerbare 4-model.

This thesis presents Wronskian and Gram solutions to both the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which are then scalable to arbitrarily large numbers of interacting solitons. Through variable transformation and use of the Hirota derivative, these nonlinear partial differential equations can be expressed in bilinear form. We present both Wronskian and Gram determinants which satisfy the equations. N=1,2,3 and higher order solutions are presented graphically; parameter tuning and the resultant behavioral differences are demonstrated and discussed. In addition, we compare these solutions to naturally occurring shallow water waves on beaches.

The ability of a ferromagnet to maintain its magnetic state in the absence of an external magnetic field has made ferromagnetic materials an important subject of study in physics since the end of the 19th century. Moreover, ferromagnetic materials are the cornerstone for data storage systems such as magnetic tapes, magnetic disk drives and magnetic random access memory. The discovery of the Giant Magneto Resistance (GMR) in 1988 suggested that, since the magnetic state of the electrical conductor has an important effect upon the current flow, there may also be an inverse influence of the current upon the magnetization. In this effect, predicted in 1989 [1] by Slonczewski and called Spin Transfer Torque, angular momentum transferred by a spin polarized current can exert a torque on the magnetization of a ferromagnetic material, changing the local magnetization and stimulating the precession of the magnetic moments, generating microwave signals. This provides a new method of manipulating magnetization without applying an external field. Large polarized currents lead to spin transfer effects which are the driving force for the magnetic dynamics of devices known as Spin Transfer Oscillators (STO). In this new kind of nano-device the emission of microwaves is stimulated by a DC electrical current and measured as a change in the output voltage due the GMR effect. The specific characteristics of these devices such as working frequency and DC current ranges, microwave emission linewidth, and maximum emission power among others, are given by the design and size of the device,and the nature of the magnetic oscillations producing the emission. Among the multiple types of STO that now exist , I have focused my research upon three of them: Spin Transfer Vortex Oscillators (STVO), Single Layer Spin Transfer Oscillators (SL-STO) and Orthogonal Pseudo Spin Valves. Within STVOs and SL-STOs we can nucleate what is called a magnetic vortex. A magnetic vortex is a curling of the in-plane of a magnetic layer with its centre pointing out of the magnetization plane. The gyration of this vortex due to STT produces a microwave emission < 1GHz with a greater emission power than that produced by the precession of magnetic moments in STOs. The phase-locked synchronisation of multiple vortices is expected to exhibit enhanced microwaved power and phase stability compared to a single vortex device, providing a solution to the drawbacks of the STO in the low frequency regime. On the other hand, Orthogonal Pseudo Spin Valves promote the nucleation of magnetic dissipative solitons, also called magnetic droplets. This type of magnetic structure has an opposite out of plane magnetization to the layer that contains it. Compared to the microwave emission of magnetic vortices , magnetic droplets have a higher frequency range and emission power. However, their nucleation is subject to large external fields being applied to the sample. In this thesis, I electrically characterized these devices and applied magnetic imaging techniques in order to go further in the understanding of the spatial features and dynamic behaviour of these magnetic structures. It is not possible to acquire this knowledge by only using electrical characterization. Understanding the magnetization dynamics in these devices is crucial for the design of STO based devices while imaging studies are required to prove the existence of these magnetic structures, as in case of the magnetic droplet. In chapter 2 I will introduce the background concepts of magnetism that are relevant to this thesis. I will go from the basics principles of ferromagnetism, its quantum mechanical treatment, and the theory that explain the dynamics of the magnetisation. I will also present the state of the art in experimental research in the field of spin transfer oscillators. My aim is to give the basic background needed to understand the results presented in this thesis. In chapter 3 I will introduce the two main experimental techniques used for imaging the magnetisation of the devices presented: Holography with Extended Reference by Autocorrelation Linear Differential Operator (HERALDO) and Time Resolved Scanning Kerr Microscopy (TRSKM). I will revise the theoretical background concepts and the development of the techniques in order to demostrate the uniqueness of each technique and how they were used in this thesis. It is interesting to note that while MOKE is a well-known and widely-used technique, far fewer laboratories in the world area able to perform time resolved measurements using MOKE, with the University of Exeter being one of them. Furthermore, HERALDO is a novel technique that is used for the first time to image magnetic structures within multilayer systems in this thesis, which is a milestone in the development of the techinque. In chapter 4 I present an investigation of the magnetization dynamics of a SL-STO. Electrical transport measurements provided an initial characterization of the device. We then used HERALDO for the first time to investigate the magnetization dynamics in an intermediate layer of a multilayer stack. We present time averaged measurements of the magnetisation of a magnetic vortex formed underneath a nano contact (NC) positioned on top of the multilayer, using a combination of x-ray holography and x-ray magnetic circular dichroism. In chapter 5 I present the first direct measurement at the time of a magnetic dissipative droplet, using holography with extended reference autocorrelation by linear differential operator (HERALDO). I studied the out of plane magnetisation of the free layer under a NC within an orthogonal pseudo spin salve. In chapter 6 I present and study STVO devices with pairs of NCs of 100 nm diameter and centre-to-centre separation D = 200 to 1100 nm, by a combination of electrical measurements and time-resolved scanning Kerr microscopy (TRSKM). It will be shown that the dynamic behaviour of vortices and anti vortices changes when the distances between the NCs within the devices is changed.

Condensados de Bose-Einstein (BEC) são sistemas macroscópicos excelentes para a observação do comportamento quântico da matéria. Desde sua obtenção experimental em gases atômicos alcalinos diluídos aprisionados por campos magnéticos, há importantes aspectos relacionados a esse sistema que foram intensamente explorados, como os modos coletivos do BEC harmonicamente aprisionado, seu tunelamento através de barreiras de potencial e os estados excitados desse sistema, incluindo vórtice e sóliton. O último consiste de pacote de onda localizado, que propaga sem mudança de forma. Nesse trabalho, investigamos os novos aspectos que surgem da dinâmica de um sistema composto (condensado aprisionado contendo um sóliton). Há muitos estudos tratando cada parte separadamente: estado fundamental do BEC ou um sóliton em um BEC infinito uniforme estacionário. Estamos nos baseando nessas análises prévias, além da simulação numérica de campo médio do nosso sistema submetido a diferentes condições iniciais (BEC aprisionado no mínimo do potencial harmônico ou BEC deslocado na armadilha contendo um sóliton, além de uma deformação no potencial) para caracterizar a dinâmica desse sistema. Alguns dos nossos resultados puderam ser explicados por meio de predições analítica da chamada aproximação de Thomas-Fermi. Ao final, comparamos as simulações de campo médio (equação de Gross-Pitaevskii) com as advindas da teoria de múltiplos orbitais a fim de justificar o regime de validade da nossa teoria. / Bose-Einstein Condensates (BEC) are excellent macroscopic systems to observe the quantum behavior of matter. Since it experimental production in dilute atomic alkali gases trapped by magnetic fields, there are important aspects related to this system that have been intensely explored, like the collective modes of the harmonically trapped BEC, its tunneling through a potential barrier and the excited states of this system, that include the vortex and soliton. The latter consist of localized disturbances, which propagate without change of form. In this work, we investigate the singular aspects that coming from the dynamics of a composite system (trapped BEC containing a soliton). There are many studies that treat each part separately, that include a fundamental state BEC or a soliton inside a uniform infinite extent stationary BEC. We are basing on these previous analyses, besides mean-field numeric simulating our particular system submitted to diferent initial conditions (minimum harmonic potential trapped BEC or dislocated trapped BEC plus a soliton, in addition to a deformation in the potential) to characterize the tunneling dynamics. Some of our results could be explained using analytical predictions of the so called Thomas-Fermi approximation. At the end, we compar the meanfield simulations (Gross-Pitavskii equation) with the simulations from the multiple orbitals theory to justify the validity regime of our theory.

nÃo hà / Nosso trabalho apresenta uma anÃlise numÃrica da propagaÃÃo e comutaÃÃo de solitons fundamentais em acopladores de dois nÃcleos de fibras nÃo lineares construÃdos com fibras de perfis de dispersÃo (DPF). NÃs mostramos, que o formato apropriado do perfil DPF à muito eficaz na recuperaÃÃo, quase total, do comportamento da comutaÃÃo do sinal original associado à situaÃÃo sem perdas. O estudo da comutaÃÃo de solitons em acopladores nÃo lineares de fibras DPF proporciona possibilidades de se alcanÃar alta eficiencia no processamento de sinais ultra-rÃpidos em sistemas totalmente Ãticos, especialmente para comutadores e transistors Ãticos. Iniciamos, estudando as equaÃÃes de Maxwell, os Solitons Ãpticos, as EquaÃÃes dos Modos Acoplados, e a EquaÃÃo NÃo Linear de SchrÃdinger - NLSE (mÃtodos analÃtico, e numÃrico). A seguir, detalhamos a PropagaÃÃo e Chaveamento de Sinais Ãpticos âCWâ e âQuasi-CWâ em Acopladores Ãpticos, bem como analisamos de forma teÃrica os acopladores duplos nÃo lineares de Fibras Ãpticas com Perfis de DispersÃo Decrescente (DPF). Os Ãltimos capÃtulos tratam do objetivo real de nossa DissertaÃÃo, ou seja, a AnÃlise Detalhada de Acopladores Ãpticos de Fibras com Perfis de DispersÃo (DPF), Com, e Sem Perdas Ãpticas, incluindo dois tipos de configuraÃÃo(PA1 e PB1). Encerramos a DissertaÃÃo com as ConclusÃes sobre os estudos acima referidos / We present a numerical investigation of the propagation and the switching of fundamental solitons in a two-core nonlinear fiber coupler constructed with Dispersion Profile Fiber (DPF). We have shown that appropriate shaping of the DPF profile is quite effective to recover, almost completely, the original switching behavior associated to the lossless situation. The study of soliton switching in DPF nonlinear fiber couplers provides possibilities for achieving, high efficiency in ultrafast all-optical signal processing, especially for optical switches and optical transistors. We initiate, studying the Maxwell Equations, the Coupled-Mode Equations, and the the Nonlinear SchrÃdinger Equation - NLSE, (analytic, and numerical approaches). After this, we detail the Propagation and Switching of "CW" and "Quasi-CW" Optical Signals in Optical Couplers, as well like we analyze, through theoretical form, the nonlinear double couplers of Dispersion Profiles Fibers (DPF). The last chapters explain the real objective of our Dissertation, or be, Detailed Analysis of the Optical Couplers of Profiles Dispersion Fiber (DPF), With, and Without Optical Losses, including two kinds of configuration (PA1 and PB1). We enclose the Dissertation with the Conclusions about the above referred studies

In this report, we study various nonlinear wave equations arising in mathematical physics and investigate the existence of solutions to these equations using variational methods. In particular, we look for particle-like traveling wave solutions known as solitary waves. This study is motivated by the prevalence of solitary waves in applications and the rich mathematical structure of the nonlinear wave equations from which they arise. We focus on a semilinear perturbation of Maxwell's equations and the nonlinear Klein - Gordon equation coupled with Maxwell's equations. Physical ramifications of these equations are also discussed.

We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of the original enlarged spectral problem generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original integrable couplings of a generalized D-Kaup-Newell soliton hierarchy to furnish its Hamiltonian structures. Then we apply the variational identity to the reduced integrable couplings. The reduced coupling system has a bi-Hamiltonian structure. Both integrable coupling systems retain the properties of infinitely many commuting high-order symmetries and conserved densities of their original subsystems and, again, are Liouville integrable. In order to find solutions to a generalized D-Kaup-Newell integrable coupling system, a theory of Darboux transformations on integrable couplings is formulated. The theory pertains to a spectral problem where the spectral matrix is a polynomial in lambda of any order. An application to a generalized D-Kaup-Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Precise one-soliton-like solutions are given for the m-th order generalized D-Kaup-Newell integrable coupling system.

L'objectif de cette thèse a été l'évaluation de deux semi-conducteurs (l'InP :Fe et le SPS :Te) comme matériaux pour les télécommunications optiques aux longueurs d'ondes infrarouges. <br /> D'abord, en ce qui concerne l'InP :Fe: nous avons fait une caractérisation systématique du phénomène d'autofocalisation photoréfractive, prenant en compte les paramètres les plus importants qui intervient dans ce phénomène (température, dopage, intensité du faisceau et de l'éclairage de fond, polarisation du faisceau). Ainsi, nous sommes maintenant capables de contrôler le phénomène d'autofocalisation. En tenant compte également des temps de réponse mesurés et des simulations réalisées, nous croyons que l'interaction de deux faisceaux autofocalisés est possible et maîtrisable sur une échelle de temps de l'ordre de microsecondes. Néanmoins, alors que nous connaissons l'influence des paramètres mis en jeu sur l'autofocalisation, le développement d'un modèle théorique reste indispensable pour une compréhension des mécanismes physiques qui déterminent la dynamique de l'autofocalisation photoréfractive. Nos mesures expérimentales et simulations théoriques ont montré que les modèles existants ne décrivent pas d'une manière satisfaisante les phénomènes observés dans InP :Fe.<br /> En revanche, l'autofocalisation observée dans le SPS :Te est décrite par les modèles "classiques" existants. On peut dire que ce deux matériaux sont complémentaires: alors que dans le SPS :Te l'autofocalisation est plus lente que dans l'InPFe, elle est plus forte et plus facile à maîtriser. Tenant compte de cette remarque, nous croyons que ces deux matériaux trouveront leur place dans de futures applications.

Le travail présenté dans ce mémoire concerne l'étude dans les microrésonateurs à semiconducteur III-V des phénomènes d'Optique Transverse que sont les solitons de cavité. Ceux-ci résultent de l'interaction d'un faisceau<br />externe injecté dans la cavité avec les non-linéarités du matériau actif et la diffraction. L'originalité de ce travail tient au pompage optique de l'amplificateur semiconducteur à cavité verticale (VCSOA). On s'est d'abord attaché à développer les aspects théoriques de la physique mise en jeu, puis les aspects pratiques et technologiques concernant la conception et la réalisation des VCSOAs présentant des caractéristiques favorables à l'émergence de phénomènes d'optique transverse. Une grande partie du travail, que ce soit théorique ou<br />technologique, concerne la maitrîse du chauffage résiduel et l'atténuation de ses effets inhibiteurs de l'observation des solitons de cavité. Les expériences présentées ici montrent pour la première fois la formation de motifs transverses et la possibilité d'écrire et d'effacer de manière cohérente et incohérente des solitons de cavité en régime de pompage optique.

Dans ce travail de thèse nous avons exposé une nouvelle méthode simple et efficace basée sur l'autofocalisation de faisceaux qui permet l'induction de guides optiques au cœur de matériaux photoréfractifs, sans tension appliquée. Cette technique de photo-inscription exploite l'effet pyroélectrique des matériaux ferroélectriques qui permet de générer un effet non linéaire focalisant contrôlé par une augmentation de quelques degrés de la température du matériau. La démonstration expérimentale est réalisée dans les échantillons de LiNbO³ non dopé. Une analyse détaillée de la dynamique de focalisation a notamment permis de former des solitons spatiaux dénommés " pyrolitons ". Ces solitons brillants permettent d'induire des guides circulaires à faibles pertes qui restent mémorisés sur de longues périodes dans le matériau. Par ailleurs, des solitons de surface se propageant à l'interface air/LiNbO³ ont également été observés grâce à cet effet non linéaire contrôlé par la pyroélectricité. Ces résultats expérimentaux ont été confirmés par un modèle numérique 3-D qui prend en compte la dynamique de l'effet photoréfractif. La non linéarité optique commandée en température révélée dans ce travail peut faciliter la mise en évidence de nouveaux concepts. En outre, la technique de photo-inscription associée ouvre de nouvelles perspectives pour la fabrication de circuits optiques 3-D dans le LiNbO³, matériau de référence pour la photonique.