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Cette thèse à pour objet l'étude théorique et numérique de solutions dans les équations aux dérivées partielles non linéaires de la physique, en particulier en dynamique des fluides. La présence de discontinuités dans les solutions de ces équations complique la compréhension mathématique des phénomènes mis enjeu et leur traitement numérique, notamment en vue de simulations informatiques . Nous étudions ces équations par une méthode de régularisation dans un espace fonctionnel approprié. Lorsque des schémas numériques construits par des méthodes différentes conduisent à des résultats identiques, ceci jusque dans leurs moindres détails, il semble alors naturel de s'interroger dans quelle mesure ces suites de solutions numériques constituent une approximation d'une solution des équations étudiées. Nous construisons des suites de solutions approchées à partir d'un schéma numérique original,stable et suffisamment simple pour démontrer que ses suites constituent une méthode asymptotique de Maslov au sens des distributions en dimension trois d'espèce. La technique de régularisation employée consiste à étendre les variables réelles du problème ne des variables complexes, ce qui nous permet de construire des familles de solutions particulières que l'on ramène au cas réel en faisant tendre un petit paramètre vers O. Les solutions physiques recherchées apparaissent alors comme valeurs au bord de fonction holomorphes. Nous illustrons les résultats obtenus par des applications en cosmologie dans les cadres Newtoniens et relativistes pour des systèmes sans pression, puis avec pression et auto-gravitation, ainsi que pour le système des gaz parfaits. / This thesis is devoted to the theoretical and numerical study of singular solutions appearing in nonlinear partial differential complicates the mathematical understanding of the phenomena under concem as well as their numerical treatment, in particular in view of computation. These equations are studied by a regularization method in an appropriate functional space. When completely different numerical methods give the same results up to the smallest details one can reasonably expect that these numerical results suggest the existence of a mathematical solution of theses equations. We construct sequences of approximate solutions from an original numerical scheme, which is stable and simple enough to prove that these sequences constitute a Maslov asymptotic method in three space dimension. The regularization technique in use consits in extending the real variables of the problem into complex ones, which perrnits to construct families of particular equations that we bring back to the real case by letting a small paramater tend to zero. The expected physical solutions appear as boundary values of holomorphie functions . Illustrations are given by applications to cosmology in the Newtorian and re1ativistic settings for pressure1ess fluid dynamics, then in presence of self-gravitation and pressure as weil as for the systemof ideal gases

Our dynamic Sun manifests its activity by different phenomena: from the 11-year cyclic sunspot pattern to the unpredictable and violent explosions in the case of solar flares. During flares, a huge amount of the stored magnetic energy is suddenly released and a substantial part of this energy is carried by the energetic electrons, considered to be the source of the nonthermal radio and X-ray radiation. One of the most important and still open question in solar physics is how the electrons are accelerated up to high energies within (the observed in the radio emission) short time scales. Because the acceleration site is extremely small in spatial extent as well (compared to the solar radius), the electron acceleration is regarded as a local process. The search for localized wave structures in the solar corona that are able to accelerate electrons together with the theoretical and numerical description of the conditions and requirements for this process, is the aim of the dissertation. Two models of electron acceleration in the solar corona are proposed in the dissertation: I. Electron acceleration due to the solar jet interaction with the background coronal plasma (the jet--plasma interaction) A jet is formed when the newly reconnected and highly curved magnetic field lines are relaxed by shooting plasma away from the reconnection site. Such jets, as observed in soft X-rays with the Yohkoh satellite, are spatially and temporally associated with beams of nonthermal electrons (in terms of the so-called type III metric radio bursts) propagating through the corona. A model that attempts to give an explanation for such observational facts is developed here. Initially, the interaction of such jets with the background plasma leads to an (ion-acoustic) instability associated with growing of electrostatic fluctuations in time for certain range of the jet initial velocity. During this process, any test electron that happen to feel this electrostatic wave field is drawn to co-move with the wave, gaining energy from it. When the jet speed has a value greater or lower than the one, required by the instability range, such wave excitation cannot be sustained and the process of electron energization (acceleration and/or heating) ceases. Hence, the electrons can propagate further in the corona and be detected as type III radio burst, for example. II. Electron acceleration due to attached whistler waves in the upstream region of coronal shocks (the electron--whistler--shock interaction) Coronal shocks are also able to accelerate electrons, as observed by the so-called type II metric radio bursts (the radio signature of a shock wave in the corona). From in-situ observations in space, e.g., at shocks related to co-rotating interaction regions, it is known that nonthermal electrons are produced preferably at shocks with attached whistler wave packets in their upstream regions. Motivated by these observations and assuming that the physical processes at shocks are the same in the corona as in the interplanetary medium, a new model of electron acceleration at coronal shocks is presented in the dissertation, where the electrons are accelerated by their interaction with such whistlers. The protons inflowing toward the shock are reflected there by nearly conserving their magnetic moment, so that they get a substantial velocity gain in the case of a quasi-perpendicular shock geometry, i.e, the angle between the shock normal and the upstream magnetic field is in the range 50--80 degrees. The so-accelerated protons are able to excite whistler waves in a certain frequency range in the upstream region. When these whistlers (comprising the localized wave structure in this case) are formed, only the incoming electrons are now able to interact resonantly with them. But only a part of these electrons fulfill the the electron--whistler wave resonance condition. Due to such resonant interaction (i.e., of these electrons with the whistlers), the electrons are accelerated in the electric and magnetic wave field within just several whistler periods. While gaining energy from the whistler wave field, the electrons reach the shock front and, subsequently, a major part of them are reflected back into the upstream region, since the shock accompanied with a jump of the magnetic field acts as a magnetic mirror. Co-moving with the whistlers now, the reflected electrons are out of resonance and hence can propagate undisturbed into the far upstream region, where they are detected in terms of type II metric radio bursts. In summary, the kinetic energy of protons is transfered into electrons by the action of localized wave structures in both cases, i.e., at jets outflowing from the magnetic reconnection site and at shock waves in the corona. / Die Sonne ist ein aktiver Stern, was sich nicht nur in den allseits bekannten Sonnenflecken, sondern auch in Flares manifestiert. Während Flares wird eine große Menge gespeicherter, magnetischer Energie in einer kurzen Zeit von einigen Sekunden bis zu wenigen Stunden in der Sonnenkorona freigesetzt. Dabei werden u.a. energiereiche Elektronen erzeugt, die ihrerseits nichtthermische Radio- und Röntgenstrahlung, wie sie z.B. am Observatorium für solare Radioastronomie des Astrophysikalischen Instituts Potsdam (AIP) in Tremsdorf und durch den NASA-Satelliten RHESSI beobachtet werden, erzeugen. Da diese Elektronen einen beträchtlichen Anteil der beim Flare freigesetzten Energie tragen, ist die Frage, wie Elektronen in kurzer Zeit auf hohe Energien in der Sonnenkorona beschleunigt werden, von generellem astrophysikalischen Interesse, da solche Prozesse auch in anderen Sternatmosphären und kosmischen Objekten, wie z.B. Supernova-Überresten, stattfinden. In der vorliegenden Dissertation wird die Elektronenbeschleunigung an lokalen Wellenstrukturen im Plasma der Sonnenkorona untersucht. Solche Wellen treten in der Umgebung der magnetischen Rekonnektion, die als ein wichtiger Auslöser von Flares angesehen wird, und in der Nähe von Stoßwellen, die infolge von Flares erzeugt werden, auf. Generell werden die Elektronen als Testteilchen behandelt. Sie werden durch ihre Wechselwirkung mit den elektrischen und magnetischen Feldern, die mit den Plasmawellen verbunden sind, beschleunigt. Infolge der magnetischen Rekonnektion als Grundlage des Flares werden starke Plasmaströmungen (sogenannte Jets) erzeugt. Solche Jets werden im Licht der weichen Röntgenstrahlung, wie z.B. durch den japanischen Satelliten YOHKOH, beobachtet. Mit solchen Jets sind solare Typ III Radiobursts als Signaturen von energiereichen Elektronenstrahlen in der Sonnenkorona verbunden. Durch die Wechselwirkung eines Jets mit dem umgebenden Plasma werden lokal elektrische Felder erzeugt, die ihrerseits Elektronen beschleunigen können. Dieses hier vorgestellte Szenarium kann sehr gut die Röntgen- und Radiobeobachtungen von Jets und den damit verbundenen Elektronenstrahlen erklären. An koronalen Stoßwellen, die infolge Flares entstehen, werden Elektronen beschleunigt, deren Signatur man in der solaren Radiostrahlung in Form von sogenannten Typ II Bursts beobachten kann. Stoßwellen in kosmischen Plasmen können mit Whistlerwellen (ein spezieller Typ von Plasmawellen) verbunden sein. In der vorliegenden Arbeit wird ein Szenarium vorgestellt, das aufzeigt, wie solche Whistlerwellen an koronalen Stoßwellen erzeugt werden und durch ihre resonante Wechselwirkung mit den Elektronen dieselben beschleunigen. Dieser Prozess ist effizienter als bisher vorgeschlagene Mechanismen und kann deshalb auch auf andere Stoßwellen im Kosmos, wie z.B. an Supernova-Überresten, zur Erklärung der dort erzeugten Radio- und Röntgenstrahlung dienen.

The present study investigates longshore sediment transport for a variety of bathymetric and wave conditions using the National Oceanic Partnership Program (NOPP) NearCoM Model. The model is used to determine the effects of wave shape and bathymetry changes on the resulting longshore sediment transport. The wave drivers, REF/DIF 1 and REF/DIF S, are used to assess the effects of monochromatic and spectral waves on longshore sediment transport, respectively. SHORECIRC is used as the circulation module and four different sediment transport models are used. Longshore transport comparisons are made with and without skewed orbital velocities in the shear stress and current velocities. It is found that the addition of skewed orbital velocities in shear stress and transport formulations increases longshore sediment transport by increasing time-varying effective shear stress. The addition of skewed orbital velocities greatly increases the transport due to advection by waves. The localized longshore sediment transport is calculated using a generic physics based method and formulas by Bagnold, Bailard, and Bowen, Watanabe, and Ribberink. The transport results for each scenario are compared to the total transport CERC, Kamphuis, and GENESIS formulas. The bathymetries tested include an equilibrium beach profile, cusped beach profiles, and barred beach profiles with different bar locations. The longshore transport on an equilibrium beach profile is modeled for a 0.2 mm and 0.4 mm grain size and transport is compared to the CERC formula. The longshore sediment transport for d=0.2 mm is larger than d=0.4 mm when wave power is small, but as wave power increases the transport for the larger grain size dominates. The transport is also affected by the addition of cusps and bars on an equilibrium beach profile. The barred beach is modified to compare transport between waves breaking at the bar, before the bar, and after the bar. The features affect the transport when the wave powers are small, but as wave heights increase the cusp and bar features induce little change on the longshore sediment transport.

Turing patterns have been studied for over 50 years as a pattern forming mechanism. To date the current focus has been on the reaction mechanism, with little to no emphasis on the diffusion terms. This work focuses on combining the simplest reaction mechanism possible and the use of nonlinear cross diffusion to form Turing patterns. We start by using two methods of bifurcation analysis to show that our model can form a Turing instability. A diffusion model (along with some variants) is then presented along with the results of numerical simulations. Various tests on both the numerical methods and the model are done to ensure the accuracy of the results. Finally an additional model that is closed to mass flow is introduced along with preliminary results. / vi, 55 leaves : ill. ; 29 cm.

Orientador: Jaime Angulo Pava / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T09:23:36Z (GMT). No. of bitstreams: 1 Natali_FabioMatheusAmorin_D.pdf: 1495954 bytes, checksum: 470d5cbae62b9e11fa888d6ab8e61976 (MD5) Previous issue date: 2007 / Resumo: Nesta tese estabelecemos condições suficientes para obter a estabilidade de soluções ondas Viajantes periódicas para equações de tipo KdV-type + ut +upux -- (Mu)x = 0, p ? N, com M sendo um operador pseudo-diferencial geral, porem com características especiais. Nossa abordagem é a de usar a teoria dos operadores totalmente positivos, o Teorema do Somatório de Poisson e a teoria das funções Elípticas de Jacobi. Em particular nós obtemos a estabilidade de uma família de soluções ondas viajantes periódicas para a equação de Benjamin-Ono e a equação KdV crítica. Nossas técnicas fornecem uma nova maneira para obter a existência e a estabilidade das ondas cnoidal e dnoidal associadas as equações de Korteweg-de Vries e modificada Korteweg-de Vries respectivamente. A teoria propõe o estudo de soluções ondas viajantes periódicas para outras equações diferencias parciais por exemplo, os resultados de estabilidade e instabilidade de soluções do tipo standing waves periódicas para a equação não linear de Schrödinger crítica / Abstract: In this thesis we establish su?cient conditions to obtain the stability of periodic travelling waves solutions for equations of KdV-type ut + upux -- (Mu)x = 0, p N, with M being a general pseudo-differential operator, but this operator has special characteristics. Our approach use the theory of totally positive operators, the Poisson summation theorem and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin-Ono equation and critical Korteweg-de Vries equation. Our techniques give a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated to the Korteweg-de Vries and modified Korteweg-de Vries equations respectively. The theory has prospects for the study of periodic travelling waves solutions of other partial diferential equations, for instance, the results of stability and instability of periodic standing wave solutions for the critical Schrödinger equation / Doutorado / Doutor em Matemática

Cette thèse porte sur l'étude expérimentale des ondes non-linéaires à la surface de l'eau. Premièrement, l'étude présente les mesures spatio-temporelles des ondes non-linéaires lors du passage sur une marche immergée. Celles-ci ont permis de séparer et d'analyser les diffèrent composants jusqu'au deuxième ordre. En particulier, la contribution de la tension de surface, a été mise en évidence en mesurant la longueur du battement de la deuxième harmonique. Les résultats obtenus ont été comparés à un modèle théorique multi-modal des coefficients de transmission et de réflexion. Dans la même configuration, la construction d'un bassin fermé en ajoutant un mur réfléchissant à la fin, a permis d'observer l'excitation de modes à basse fréquence, avec une dynamique quasi-périodique intéressante. En parallèle, deux aspects expérimentaux impliqués dans les manipulations à petite échelle ont été étudiés. Premièrement, l'atténuation produite par la friction sur le fond a été mesurée et analysée pour des ondes distribuées de façon aléatoire, en montrant l'importance relative de cet effet. Deuxièmement, la dynamique de la ligne de contact joue un rôle important lorsque les ondes ont des amplitudes suffisamment petites et que les bords se trouvent suffisamment proches. Dans ce cas, nous avons constaté des différences considérables en réflexion et en courbure du front d'onde. La dernière partie porte sur les mesures expérimentales de l'absorption parfaite avec un résonateur couplé dans un guide d'onde étroit. Les modes piégés générés par un cylindre décalé dans le guide, ont été excités pour produire l'absorption. / This thesis presents an experimental investigation on the propagation of nonlinear water waves. The first part focuses on the space-time measurements of nonlinear water waves, when it passes over a submerged step. The space-time resolved measurement allows us to separate the different components at the second order, which are compared with a theoretical nonlinear multi-modal model. The important contribution of the surface tension at higher orders is verified by measuring the beating length of the second harmonic. In the same conditions, the addition of a reflecting wall at the end of the channel sets a rectangular tank with submerged step, where the excitation of low-frequency modes yields a quasi-periodic dynamics. Concurrently, a research about aspects that have to be considered in small scale experiments of surface waves has been carried out. In shallow water, the damping of water waves is highly influenced by the bottom friction. This dependence was measured for randomly distributed waves, revealing the relative contribution of this effect. Moreover, the dynamic of the contact line plays a significant role when the wave-amplitude is small and the boundaries are near, both in relation to the capillary length. We observed experimentally how the wetting of the boundaries changes the reflection and the wave-front curvature. The final part covers the measurement of perfect wave absorption by a coupled resonator in a narrow waveguide. The trapped modes generated by a cylinder shifted from the channel axis were excited to generate the absorption.

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.

The focus of this dissertation is the existence, stability, and resulting dynamical evolution of localized stationary solutions to Nonlinear Schr¨odinger (NLS) equations with periodic confining potentials in 2(+1) dimensions. I will make predictions about these properties based on a discrete lattice model of coupled ordinary differential equations with the appropriate symmetry. The latter has been justified by Wannier function expansions in a so-called tight-binding approximation in the appropriate parametric regime. Numerical results for the full 2(+1)-D continuum model will be qualitatively compared with discrete model predictions as well as with nonlinear optics experiments in optically induced photonic lattices in photorefractive crystals. The predictions are also relevant for BECs (Bose-Einstein Condensates) in optical lattices.

<p>Ferrofluids are remarkable materials composed of magnetic nanoparticles dispersed in a carrier liquid. These suspensions exhibit fluid-like behavior in the absence of a magnetic field, but when exposed to a magnetic field, they can respond and deform into a variety of patterns. This responsive behavior of ferrofluids makes them an excellent material for applications such as drug delivery for targeted therapies and soft robots. In this thesis, we will focus on the interfacial dynamics of ferrofluids in Hele-Shaw cells. The three major objectives of this thesis are: understanding the pattern evolution, unraveling the underlying nonlinear dynamics, and ultimately achieving passive control of ferrofluid interfaces. First, we introduce a novel static magnetic field setup, under which a confined circular ferrofluid droplet will deform and spin steadily like a `gear’, driven by interfacial traveling waves. This study combines sharp-interface numerical simulations with weakly nonlinear theory to explain the wave propagation. Then, to better understand these interfacial traveling waves, we derive a long-wave equation for a ferrofluid thin film subject to an angled magnetic field. Interestingly, the long-wave equation derived, which is a new type of generalized Kuramoto--Sivashinsky equation (KSE), exhibits nonlinear periodic waves as dissipative solitons and reveals fascinating issues about linearly unstable but nonlinearly stable structures, such as transitions between different nonlinear periodic wave states. Next, inspired by the low-dimensional property of the KSE, we simplify the original 2D nonlocal droplet problem using the center manifold method, reducing the shape evolution to an amplitude equation (a single local ODE). We show that the formation of the rotating `gear’ arises from a Hopf bifurcation, which further inspires our work on time-dependent control. By introducing a slowly time-varying magnetic field, we propose strategies to effectively control a ferrofluid droplet's evolution into a targeted shape at a targeted time. The final chapter of this thesis concerns our ongoing research into the interfacial dynamics under the influence of a fast time-varying and rotating magnetic field, which induces a nonsymmetric viscous stress tensor in the ferrofluid, requiring the balance of the angular momentum equation. As a consequence, wave propagation on a ferrofluid interface can be now triggered by magnetic torque. A new thin-film long-wave equation is consistently derived taking magnetic torque into account.</p>