Nonlinear, nonlocal optical media has emerged as an ideal setting for experimentally observing and studying spatial optical solitary waves which otherwise cannot occur in Kerr media. Of particular interest is the eventual application to all-optical circuits. However, there is considerable work left to do on the theoretical end before this is a possibility. In this thesis we consider three problems. The first is how to solve the governing equations for optical beam propagation in the particular medium of the nematic liquid crystal (NLC), which is used as a prototypical example, exactly and approximately. In this respect we provide the first known, explicit solutions to the model as well as a comprehensive assessment on how to use variational, or modulation theory, in this context. This leads to the discovery of a novel form of bistability in the system, which shows there are two stable solitary wave solutions for a fixed power or L2 norm. We then consider how to approximate solutions for optical solitary waves propagating in a more general class of nonlocal nonlinear media using asymptotic methods. This is a long open problem and is resolved in the form of a simple to implement method with excellent accuracy and general applicability to previously intractable models. We conclude with the discovery and characterization of an instability mechanism in a coupled, defocussing nonlinear Schrodinger system. We show there is no stable, coupled, localized solution. This result is compared with the more well-studied bright solitary wave system and physical and mathematical explanations are offered.

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.

Au moyen d'une expérience d'excitation paramétrique d'onde de surface, nous mettons en évidence l'existence de nouveaux types d'ondes solitaires et stationnaires à la surface de l'eau. Ces ondes de grande amplitude sont très non-linéaires et l'étude théorique réalisée ne permet pas de rendre compte de la forme des vagues mais permet de comprendre l'origine du phénomène d'hystérésis observé qui est nécessaire à la compréhension des phénomènes observés. En effet, l'existence de ces ondes (dans notre configuration expérimentale) est conditionnée par la présence d'un domaine de bistabilité dans le plan amplitude d'excitation - amplitude des vagues au coeur duquel nous avons montré qu'il était possible d'avoir coexistence de deux solutions, une d'amplitude nulle et une d'amplitude non nulle. Ces expériences en géométrie Hele-Shaw ont aussi permis de mettre en évidence des ondes enveloppes qui ne sont encore décrit par aucun modèle existant. Il s'agit à notre connaissance de la première onde enveloppe stationnaire observé à la surface de l'eau. Nous mettons aussi en évidence des ondes de gravité de très grande amplitude, qui sont formées alternativement d'étoiles et de polygones. Nous montrons que la symétrie du motif (nombre de branche de l'étoile) est indépendante de la taille et de la forme du récipient vibré. Nous montrons qu'un mécanisme de couplage non-linéaire résonant à trois ondes peut expliquer cette géométrie, bien que cette possibilité fut rejetée pour des ondes purement gravitaire. / By means of the parametric excitation of water waves in a Hele-Shaw cell, we report the existence of two new types of highly localized, standing surface waves of large amplitude. They are respectively of odd and even symmetries. Both solitary waves oscillate subharmonically with the forcing frequency. They are highly nonlinear, and dier strongly from the other types of localized patterns. Moreover, to our knowledge, such a solitary waves of odd symmetry has never been reported hitherto. We report a new type of standing gravity waves of large amplitude, having alternatively the shape of a star and of a polygon. This wave is observed by means of a laboratory experiment by vibrating vertically a tank. The symmetry of the star (i.e. the number of branches) is independent of the container form and size, and can be changed according to the amplitude and frequency of the vibration. We show that this wave geometry results from nonlinear resonant couplings between three waves, although this possibility has been denied for pure gravity waves up to now.

The breaking of internal solitary waves (ISWs) of depression shoaling upon a uniformly sloping boundary in a smoothed two-layer density field was investigated using high-resolution two-dimensional simulations. The simulations were performed for a wide range of boundary slopes S∈[0.01,0.3] and wave slopes. Over steep slopes (S≥0.1), three distinct breaking processes were observed; surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary layer separation, respectively. Over mild slopes (S≤0.05), nonlinearity varies gradually and the wave fissions into a train of waves of elevation after it passes through the turning point where solitary waves reverse polarity. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relative developmental timescale. The breaker type was characterized in wave slope S_w versus S space and the reflection coefficient (R), modeled as a function of the internal Iribarren number, was in agreement with other studies. The same 2D model was applied to investigate boundary layer separation-driven global instability, which is shown to play an important role in breaking of shoaling ISWs. The simulations were conducted with waves propagating over a flat bottom and shoaling over relatively mild (S=0.05) and steep (S=0.1) slopes. Combining the results over flat and sloping boundaries, a unified criterion for vortex shedding is proposed, which depends on the momentum thickness Reynolds number and the non-dimensionalized ISW-induced pressure gradient at the point of separation. The criterion is generalized to a form that may be readily computed from field data and compared to published laboratory and field observations. During vortex shedding, the bed shear stress, vertical velocity and near-bed Reynolds stress were elevated, implying potential for sediment re-suspension. Laboratory experiments were also performed to study three-dimensionality (3D) of global instability. Our results agree with previous laboratory experiments, using the proposed pressure gradient parameter and Reynolds number. The 3D effects prevent the vortices from ascending as high as they do in 2D simulations. The instabilities were not able to re-suspend sediments with 20 µm median diameters, which suggests applying lighter sediments, as finer sediments will be cohesive and dynamically different. / Thesis (Ph.D, Civil Engineering) -- Queen's University, 2011-12-23 15:03:29.76

Les ondes internes sont observées partout dans le monde. Elles ont un rôle important dans la mise en place de la chaine trophique, et elles peuvent avoir un impact sur les activités humaines. Ainsi, dans le Golfe de Guinée, des ondes internes solitaires (OIS) ont provoqué des incidents sur des plateformes pétrolières. L’objectif de la thèse est d’identifier leurs zones de génération, leurs directions de propagation, ainsi que les conditions d’environnement susceptibles de les déclencher. La thèse a comporté deux aspects : d’une part des mesures de courant étaient disponibles et d’autre part la mise en place de modèles numériques. Ces mesures ont montré le passage d’OIS. En outre, les zones de génération potentielles sont identifiées en haut du talus continental. Leur répartition mensuelle a montré une variabilité saisonnière, due à la Mousson Ouest Africaine qui modifie les conditions de stratification au cours de l’année.La modélisation a été réalisée avec HYCOM. Une maquette hydrostatique régionale du Golfe de Guinée a été mise en place. Elle a d’abord été validée en configuration barotrope (océan homogène). Une configuration bicouche a ensuite été testée, et la théorie linéaire des ondes internes a été vérifiée sur ses résultats. Des termes non hydrostatiques ont ensuite été ajoutés dans les équations horizontales du courant de HYCOM, pour modéliser des OIS. L’impact des paramètres physiques, en particulier la force de Coriolis et le forçage barotrope, ainsi que l’effet d’un courant moyen, a été étudié sur des configurations bicouches radiales. Enfin, un canyon a été ajouté au milieu du talus pour évaluer qualitativement les effets d’une bathymétrie 2D sur les OIS. / Internal waves are observed in different locations. They have a key role in the set up of the trophic chain, and they can impact human activities.For instance, in the Gulf of Guinea, internal solitary waves (ISWs) have caused hazards on offshore oil platforms. The aim of the thesis is to identify the generation zones and the directions of propagation of these ISWs, and the environmental conditions (tides, stratification) that can generate them. One the one hand measurements of currents were available, and on the other hand numerical models were used.Measurements of currents were analyzed. Packets of ISWs were identified. Moreover, the potential generation zones of the ISWs, particularly the top of the shelf break, were inferred from their direction of propagation. The monthly repartition of the packets of ISWs revealed a seasonal variability, due to the West African Monsoon, which modified the conditions of the stratification throughout the year.Concerning the modelling, the numerical model HYCOM was used. One regional hydrostatic model of the Gulf of Guinea was built. This regional model was validated for a barotropic configuration (homogeneous ocean) first. Then, a two-layered configuration was set up, and the linear theory of the internal waves was confirmed on the results.Non hydrostatic terms were added to the horizontal velocity equations in HYCOM, to model ISWs. The influence of different physical parameters, such as the Coriolis force, the barotropic forcing, and a mean current was studied on two-layered radial configurations. Finally, a canyon in the middle of the shelf break was set up to assess qualitatively the effect of a 2D bathymetry on the ISWs.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-02Bitstream added on 2014-06-13T19:32:40Z : No. of bitstreams: 1 carvalho_pgs_me_ift.pdf: 273148 bytes, checksum: b8f03ddb09e18a5f045875f4ec5b3d33 (MD5) / Estuda-se o comportamento de ondas de superfície em um fluido invícido no limite de curtos comprimentos de onda e levando em conta efeitos não-lineares. Uma equação para este limite é obtida e algumas soluções são exibidas. / Abstracts: The behaviour of surface-waves on a inviscid fluid is studied in the short-wave limit, taking nonlinear effects into account. An equation describing this limit is obtained and some of its solutions exibited.

In this paper we construct explicit examples of solutions to certain nonlinear wave equations. These semilinear equations are the simplest equations known to possess localized solitary waves in more that one spatial dimension. We construct explicit localized standing wave solutions, which generate multidimensional localized traveling solitary waves under the action of velocity boosts. We study the case of two spatial dimensions and a piecewise-linear nonlinearity. We obtain a large subset of the infinite family of standing waves, and we exhibit several interesting features of the family. Our solutions include solitary waves that carry nonzero angular momenta in their rest frames. The spatial profiles of these solutions also furnish examples of symmetry breaking for nonlinear elliptic equations.

Orientador: Roberto André Kraenkel / Banca: Gerson Francisco / Banca: Samuel Maier Kurcbart / Resumo: Estuda-se o comportamento de ondas de superfície em um fluido invícido no limite de curtos comprimentos de onda e levando em conta efeitos não-lineares. Uma equação para este limite é obtida e algumas soluções são exibidas. / Abstracts: The behaviour of surface-waves on a inviscid fluid is studied in the short-wave limit, taking nonlinear effects into account. An equation describing this limit is obtained and some of its solutions exibited. / Mestre

<div>Lattices composed of bistable elements are of great significance across various fields of science and engineering due to their ability to support a class of solitary waves, called transition waves. Common with all solitary waves, transition waves carry highly concentrated energy with minimal degradation and thus have many useful engineering applications, such as extreme waveguides, bandgap transmission, vibration absorption, and energy harvesting. The rich dynamics arising from the strong nonlinearities of the constitutive bistable microstructures still have much to be unveiled for the practical implementation of the transition waves in real-world engineering structures. Especially, the quasi-particle characteristics of the transition waves can potentially address the performance limits posed by the unit cell size in linear metamaterials.</div><div><br></div><div>In this thesis, we first present an input-independent generation of transition waves in the lattices of asymmetric bistable unit cells when snap-through transitions occur at any site within the lattice. The resulting responses are invariant across the lattice except near the boundaries. These characteristics imply useful applications in broadband energy harvesting, exploiting the highly concentrated energy of the transition waves. We further observe that the inherent lattice discreteness induces dominantly monochromatic oscillatory tail following the main transition wave. This radiated energy of the tail can always be efficiently harvested through resonant transduction regardless of the input excitations. This type of bistable lattice transforms any input disturbance into an output form that can be conveniently transduced; thus, energy harvesting becomes an inherent metamaterial property of the bistable lattice.</div><div><br></div><div>To enhance the responses further for improved energy harvesting capability, we introduce engineered defects in the form of a mass impurity, inhomogeneous inter-site stiffness, and their combinations, achieving localization of energy at desired sites. Remarkably, we also observe a long-lived breather-like mode for the first time in this type of lattice. To enhance the tail motions globally across the lattice, we investigate the responses in a set of bistable lattices with the same mass and elastic densities but with different lattice spacing distances (or lattice discreteness). From the available tail energy, we observe a significant increase in the harvesting capability with the increased lattice discreteness.</div><div><br></div><div>Next, the effect of functional grading on the onsite and inter-site stiffnesses are investigated to augment the control of the transition waves in the bistable lattices. The unidirectionality still remains in the direction of decreasing stiffness, while a boomerang-like wave reversal occurs in the direction of increasing stiffness. Both the compression and rarefaction transition waves are allowed to propagate, enabling continuous transmission of the transition waves without complex resetting mechanisms, thus expanding the bistable lattices' functionality for practical applications.</div><div><br></div><div>The observed input-independent dynamics of the one-dimensional bistable lattices can be extended to higher-dimensional metastructures by allowing macrostructural flexibility. Metabeams composed of spring-joined bistable elements are subjected to in-plane sinusoidal input at the microstructural level, and the out-of-plane responses at the macrosctructural level are measured. As long as transition waves are triggered within the metabeam, the most dominant output frequency occurs near the natural frequency of the macroscopic structure regardless of the input excitations initiating the transition waves, yielding energy transfer between uncorrelated frequencies.</div><div><br></div><div>Finally, high-fidelity in-house numerical solvers are developed for the massively parallelized computation of the problems involving generic bistable architectures, addressing the problem size limit. The improved numerical solution accuracy and computational performance, compared to those of commercial solvers, provide great potential to discover new dynamics by drastically expanding the accessible analysis regimes.</div><div><br></div><div>The experiments, simulations, and theoretical contributions in this thesis illustrate the possibilities afforded by strongly nonlinear phenomena to tailor the dynamics of materials systems. Importantly, the presented results show mechanisms to affect global dynamic properties unconstrained by the unit cell size, thereby offering new routes to extreme dynamics beyond current metamaterial architectures.</div>

Yes / Understanding the dynamic free surface of geophysical flows has the potential to enable direct inference of the flow properties based on measurements of the free surface. An important step is to understand the inherent response of free surfaces in depth-limited flows. Here a model is presented to demonstrate that free surface oscillatory spatial correlation patterns result from individual surface features oscillating vertically as they advect over space and time. Comparison with laboratory observations shows that these oscillating surface features can be unambiguously explained by simple harmonic motion, whereby the oscillation frequency is controlled by the root-mean-square water surface fluctuation, and to a lesser extent the surface tension. This demonstrates that the observed “complex” wave pattern can be simply described as an ensemble of spatially and temporally distributed oscillons. Similarities between the oscillon frequency and estimated frequency of near-bed bursting events suggest that oscillon behaviour is linked with the creation of coherent flow structures.