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We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.

Resumo: O interesse em conhecer o comportamento supercondutor tem sido cada vez maior nas últimas décadas. Na busca de melhores características supercondutoras, descobriu-se que amostras volumétricas apresentam características muito diferentes de amostras mesoscópicas (amostras com dimensões próximas dos comprimentos de penetração de London e coerência). Como exemplo, podemos citar a não formação de rede de Abrikosov, como consequência do efeito de confinamento (efeito associado às dimensões reduzidas da amostra) e também uma mudança considerável nos valores dos campos críticos. Neste trabalho foram resolvidas as equações de Ginzburg-Landau dependentes do tempo (TDGL), para fazer uma análise detalhada da dinâmica de vórtices em filmes finos mesoscópicos. Para revolvê-las, utilizamos o método das variáveis de ligação com invariância de calibre, adaptado para o algoritmo de diferenças finitas, utilizado para obter a densidade dos pares de Cooper e também curvas de magnetização. O estudo dessa dinâmica de vórtices, foi feito em três amostras com superfícies geométricas diferentes (côncova, convexa e rugosa). Observamos que na comparação entre as duas primeiras, há uma diferença considerável nos valores dos campos críticos, bem como no comportamento da magnetização comparado com um filme plano. Já para a amostra de superfície rugosa, observamos que existe uma competição entre o efeito de confinamento e a rugosidade em relação à configuração dos vórtices. Apresentamos também, uma tabela que mostra resumidamente os estados estacionários dos vórtices nas três amostras. / Abstract: The interest to investigate the investigate the behavior of a superconductor has grown in the last few decades. Having in mind to search for better superconducting characteristics, it has been found that bulk samples present characteristics much more different than mesoscopic samples (samples with dimensions of the same order of the same order of the London penetration length and the coherence length). As an example, we can mention the non-formation of an Abrikosov vortex lattice as a consequence of the confinement effect (effect associated with the reduced dimensions of the sample) and also considerable change in the critical field values. In the present work we solved the time dependent Ginzburg-Landau equation (TDGL), in order to make a detailed analysis of the vortex dynamics in mesoscopic thin films. To solve these equations, we have used the link variables method which is gauge invariant. From this, we obtain the Cooper pair density and the magnetization curves. The vortex dynamics was investigated for three different surfaces of the film (concave, convex, and irregular). We have observed that, with respect to the parabolic geometries, there is a considerable difference for the critical fields, as well as for the behavior of the magnetization compared to a flat film. On the other hand, for a sample with an irregular surface, we have seen that there is a competition between the confinement effect and rugosity with respect to vortex configurations. We also present a table which summarizes the vortex stationary states for the three topologies mentioned above. / Orientador: Paulo Noronha Lisboa Filho / Coorientador: Edson Sardella / Banca: Wilson Aires Ortiz / Banca: Clelio Clemente de Souza Silva / Mestre

Made available in DSpace on 2014-06-11T19:30:19Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-04-28Bitstream added on 2014-06-13T19:39:47Z : No. of bitstreams: 1 pascolati_mcv_me_bauru.pdf: 4047066 bytes, checksum: cf327b8f4cd0447dc8dd5e7c4d6604a5 (MD5) / O interesse em conhecer o comportamento supercondutor tem sido cada vez maior nas últimas décadas. Na busca de melhores características supercondutoras, descobriu-se que amostras volumétricas apresentam características muito diferentes de amostras mesoscópicas (amostras com dimensões próximas dos comprimentos de penetração de London e coerência). Como exemplo, podemos citar a não formação de rede de Abrikosov, como consequência do efeito de confinamento (efeito associado às dimensões reduzidas da amostra) e também uma mudança considerável nos valores dos campos críticos. Neste trabalho foram resolvidas as equações de Ginzburg-Landau dependentes do tempo (TDGL), para fazer uma análise detalhada da dinâmica de vórtices em filmes finos mesoscópicos. Para revolvê-las, utilizamos o método das variáveis de ligação com invariância de calibre, adaptado para o algoritmo de diferenças finitas, utilizado para obter a densidade dos pares de Cooper e também curvas de magnetização. O estudo dessa dinâmica de vórtices, foi feito em três amostras com superfícies geométricas diferentes (côncova, convexa e rugosa). Observamos que na comparação entre as duas primeiras, há uma diferença considerável nos valores dos campos críticos, bem como no comportamento da magnetização comparado com um filme plano. Já para a amostra de superfície rugosa, observamos que existe uma competição entre o efeito de confinamento e a rugosidade em relação à configuração dos vórtices. Apresentamos também, uma tabela que mostra resumidamente os estados estacionários dos vórtices nas três amostras. / The interest to investigate the investigate the behavior of a superconductor has grown in the last few decades. Having in mind to search for better superconducting characteristics, it has been found that bulk samples present characteristics much more different than mesoscopic samples (samples with dimensions of the same order of the same order of the London penetration length and the coherence length). As an example, we can mention the non-formation of an Abrikosov vortex lattice as a consequence of the confinement effect (effect associated with the reduced dimensions of the sample) and also considerable change in the critical field values. In the present work we solved the time dependent Ginzburg-Landau equation (TDGL), in order to make a detailed analysis of the vortex dynamics in mesoscopic thin films. To solve these equations, we have used the link variables method which is gauge invariant. From this, we obtain the Cooper pair density and the magnetization curves. The vortex dynamics was investigated for three different surfaces of the film (concave, convex, and irregular). We have observed that, with respect to the parabolic geometries, there is a considerable difference for the critical fields, as well as for the behavior of the magnetization compared to a flat film. On the other hand, for a sample with an irregular surface, we have seen that there is a competition between the confinement effect and rugosity with respect to vortex configurations. We also present a table which summarizes the vortex stationary states for the three topologies mentioned above.

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic--quintic Ginzburg--Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non--integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse--type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this dissertation, we develop a theoretical framework for these novel classes of solutions. In the first part, we use a traveling wave reduction or a so--called spatial approximation to comprehensively investigate the bifurcations of plane wave and periodic solutions of the CGLE. The primary tools used here are Singularity Theory and Hopf bifurcation theory respectively. Generalized and degenerate Hopf bifurcations have also been considered to track the emergence of global structure such as homoclinic orbits. However, these results appear difficult to correlate to the numerical bifurcation sequences of the dissipative solitons. In the second part of this dissertation, we shift gears to focus on the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. For this part, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the two alternative starting formulations for the Lagrangian are recent and not well explored. Also, after extensive discussions with David Kaup, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler--Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well--known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors constructed via the method of multiple scales yield solitons whose amplitudes are non--stationary or time dependent. In particular, pulsating, snake (and, less easily, creeping) dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves --- their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Results will be presented for the pulsating and snake soliton cases. Chaotic evolution of the trial function parameters in chaotic regimes identified using dynamical systems analysis would yield chaotic solitary waves. The method also holds promise for detailed modeling of chaotic solitons as well. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.

In this dissertation, I study a nonlinear partial differential equation, the complex Ginzburg-Landau (CGL) equation. I first employed the perturbative field-theoretic renormalization group method to investigate the critical dynamics near the continuous non-equilibrium transition limit in this equation with additive noise. Due to the fact that time translation invariance is broken following a critical quench from a random initial configuration, an independent ``initial-slip'' exponent emerges to describe the crossover temporal window between microscopic time scales and the asymptotic long-time regime. My analytic work shows that to first order in a dimensional expansion with respect to the upper critical dimension, the extracted initial-slip exponent in the complex Ginzburg-Landau equation is identical to that of the equilibrium model A. Subsequently, I studied transient behavior in the CGL through numerical calculations. I developed my own code to numerically solve this partial differential equation on a two-dimensional square lattice with periodic boundary conditions, subject to random initial configurations. Aging phenomena are demonstrated in systems with either focusing and defocusing spiral waves, and the related aging exponents, as well as the auto-correlation exponents, are numerically determined. I also investigated nucleation processes when the system is transiting from a turbulent state to the ``frozen'' state. An extracted finite dimensionless barrier in the deep-quenched case and the exponentially decaying distribution of the nucleation times in the near-transition limit are both suggestive that the dynamical transition observed here is discontinuous. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308 / Doctor of Philosophy / The complex Ginzburg-Landau equation is one of the most studied nonlinear partial differential equation in the physics community. I study this equation using both analytical and numerical methods. First, I employed the field theory approach to extract the critical initial-slip exponent, which emerges due to the breaking of time translation symmetry and describes the intermediate temporal window between microscopic time scales and the asymptotic long-time regime. I also numerically solved this equation on a two-dimensional square lattice. I studied the scaling behavior in non-equilibrium relaxation processes in situations where defects are interactive but not subject to strong fluctuations. I observed nucleation processes when the system under goes a transition from a strongly fluctuating disordered state to the relatively stable “frozen” state where its dynamics cease. I extracted a finite dimensionless barrier for systems that are quenched deep into the frozen state regime. An exponentially decaying long tail in the nucleation time distribution is found, which suggests a discontinuous transition. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308.

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.

Ce mémoire de thèse présente une approche semi-analytique des différentes solutions solitons spatio-temporelles de l'équation cubique quintique de Ginzburg-Landau complexe étendue à (3+1)D (GL3D).La méthode semi-analytique choisie est celle des coordonnées collectives qui permet d'approcher le champ exact, dont l'expression analytique est inconnue, par une fonction d'essai, qui comporte un nombre limité de paramètres physiques.En appliquant cette procédure à l'équation GL3D, nous obtenons un système d'équations variationnelles qui gouverne l'évolution des paramètres de la balle de lumière. Nous montrons que cette approche des coordonnées collectives est incomparablement plus rapide que la procédure de résolution directe de l'équation GL3D. cette rapidité permet d'obtenir, en un temps record, une cartographie générale des comportements dynamiques des balles de lumière. Cette cartographie révèle une riche variété d'états dynamiques faite de balles de lumière stationnaires, oscillantes et rotatives.Finalement, les résultats de cette thèse prédisent l'existence de plusieurs familles de balles de lumière, et précisent les domaines respectifs de leurs paramètres physiques. Cette prédiction constitue un pas en avant dans les efforts entrepris ces dernières années en vue d'une démonstration expérimentale de ce type d'impulsions. / This thesis presents a semi-analytical approach for the search of (3+1)D spatio-temporal soliton solutions of the complex cubic-quintic Ginzburg-Landau equation (GL3D).We use a semi-analytical method called collective coordinate approach, to obtain an approximate profile of the unknown pulse field. This ansatz function is chosen to be a function of a finite number of parameters describing the light pulse.By applying this collective corrdinate procedure to the GL3D equation, we obtain a system of variational equations which give the evolution of the light bullet parameters as a function of the propagation distance. We show that the collective coordinate approach is uncomparably faster than the direct numerical simulation of the propagation equation. This permits us to obtain, efficiently, a global mapping of the dynamical behavior of light bullets, which unveils a rich variety of dynamical states comprising stationary, pulsating and rotating light bullets.Finally the existence of several types of light bullets is predicted in specific domains of the equation parameters. Altogether, this theoretical and numerical work may be a useful tool next to the efforts undertaken these last years observing light bullets experimentally.

Nous présentons une étude de la dynamique de la lumière et des mesures des caractéristiques non-linéaires optiques dans le disulfure de carbone.Dans la première partie, nous calculons dans le cadre d’un modèle classique des expressions des susceptibilités non-linéaires jusqu’au cinquième ordre, en tenant compte des corrections de champ local. Nous formulons différentes hypothèses que nous confirmons ou infirmons par la mesure des indices d’absorption et de réfraction non-linéaires. Celles-ci sont obtenues en combinant deux méthodes de caractérisation des non-linéarités au sein d’un système 4fd’imagerie. L’analyse des données expérimentales utilise une méthode nouvellement développée, qui consiste à inverser numériquement, par la méthode de Newton, les solutions analytiques des équations différentielles qui décrivent l’évolution du faisceau.Dans la deuxième partie, nous observons la filamentation d’un faisceau laser à la longueur d’onde de 532 nm et en régime picoseconde. Puis nous procédons à la mesure de l’indice de réfraction non-linéaire effectif du troisième ordre n2,eff en fonction de l’intensité incidente. Par un ajustement de la courbe de saturation de l’effet Kerr,nous développons un nouveau modèle. La résolution numérique de celui-ci reproduit la filamentation observée.La dernière partie est consacrée à l’étude de la dynamique des solitons dissipatifs au sein de milieux à gains et pertes non-linéaires. La résolution numérique de l’équation complexe de Ginzburg-Landau cubique-quintique est réalisée suivant différentes configurations :soliton fondamental, dipôle, quadrupôle,vortex carré et rhombique. / We present a study of light dynamics and measurements of the nonlinear optical characteristics of carbon disulphide. In the first part, we calculate using the classical model, the nonlinear susceptibilities up to the fifth order taking into account local field corrections. We express different assumptions that we confirm or refute by measuring the nonlinear absorption coefficient and the nonlinear refractive index. The measurements are performed by means of two nonlinear characterization methods combined with an imaging 4f system. We analyse the experimental data using a newly developed method which numerically inverts the analytical solutions of the differential equations which describe the evolution of the beam, using Newton’s method. In the second part, we observe light filamentation at wavelength 532 nm, in the picoseconds regime. Then we measure the effective third order nonlinear refractive index n2,eff versus the incident intensity. By fitting the curve of the Kerr effect saturation, we develop a new model. Numerically solving this model, allows us to reproducethe experimentally observed filamentation. The last part is dedicated to the study of dissipative solitons dynamics. The complex Ginzburg-Landau equation with cubic-quintic nonlineraties is numerically solved in various configurations : soliton fundamental dipole, quadrupole, vortex and square rhombic.

Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert.