This thesis concerns soliton equations in two spatial dimensions. Camassa and Holm derived a model of shallow water wave which continues to attract attention in the light of the remarkable wealth of mathematical and physical properties of its solutions. More recently, Kraenkel and Zenchuk introduced an extension of Camassa and Holm's equation to two spatial dimensions, with three fields, and derived it from a Lax pair. However, finding solutions of this model is more challenging. In order to understand and find solutions of the (2 + l)-dimensional Camassa- Holm (CH) equation, it turns out that we have to study and analyse1ther soliton equations in two spatial dimensions, together with their hierarchies. To begin with, we consider a squared eigenfunction symmetry of the Kadomtsev-Petviashvili (KP) hierarchy, this results in a system of equations in (2 + 1)-dimensions. One of the main tools employed is Hirota's bilinear method. Using Hirota's per- turbation technique, one-, two- and three-soliton solutions of the KP squared eigen- function flow directly constructed, and numerical plots are presented. An ansatz for multisoliton solutions is proposed and proved by induction that it is correct. The soliton formula is shown to be equivalent to the formulae found by Freeman, Gilson and Nimmo, who derived Wronskian formulae for multisoliton solutions.

No description available.

The vector nonlinear Schrödinger equation is an envelope equation which models the propagation of ultra-short light pulses and continuous-wave beams along optical fibres. Previous work has focused almost entirely on soliton solutions to the equation using a Lax representation originally developed by Manakov. We prove recursion formulae for the family of higher-order nonlinear Schrödinger equations, along with its associated Lax hierarchy, before investigating finite gap solutions using an algebrogeometric approach which introduces Baker-Akhiezer functions defined upon the Riemann surface of the relevant spectral curve. We extend this approach to account for solutions of arbitrary genus and compare it with an alternative method describing solutions of genus two. The scalar nonlinear Schrödinger and Heisenberg ferromagnet equations were shown to be equivalent following work by Lakshmanan; we generalise this idea by introducing the Heisenberg ferromagnet hierarchy and show it is entirely gauge equivalent to the scalar nonlinear Schrödinger hierarchy in the attractive case. We also investigate the polarisation state evolution of general solutions to the vector nonlinear Schrödinger equation and study possible degenerations to the Heisenberg ferromagnet equation.

In principle Quantum Mechanics allows the creation of macroscopic mass superposition states - so called "Schrödinger Cat States". This has not been confirmed experimentally largely due to the difficulty of isolating such states from environmental decoherence. It is of interest to create massive superpositions both in order to test Quantum Mechanics and to shed light on the elusive 'measurement problem'. This thesis presents the theoretical analysis of, and the initial experimental steps towards, an ambitious proposal to test the superposition principle of Quantum Mechanics at the 10^-12 kg-scale, approximately nine orders of magnitude more massive than any superposition observed to date. The experimental principle is that a small mirror mounted on a micro-mechanical oscillator (cantilever) forms one end of a high-finesse cavity in one arm of a Michelson interferometer and is coupled to a single photon by radiation pressure. The photon, in a superposition of each arm, and the cantilever evolve into a superposition involving two distinct locations of the cantilever. By observing the interference of the photon only, one can study the creation and decoherence of the combined state. Firstly, a detailed analysis of the experimental requirements is given based on (1) the need for sufficient momentum transfer from the photon to displace the micro-mirror/cantilever to a distinguishable degree, (2) the need to isolate the cantilever to avoid significant environmental decoherence, and (3) the need to have sufficient interferometric stability to perform the measurement. An iterative analysis was performed to optimise these to a set that is feasible with current technology. This demands: (1) cavity mirrors with a reflectivity of R ≥ 0.9999998 at visible wavelength, (2) a system temperature of ≤ 3mK, (3) a cantilever mechanical quality Q ≥ 10⁵ , (4) a vacuum with gas particle density of 1012/m3, (5) a relative position stability of the cavity mirrors of ≤ 10^-13 m/min, and (6) optical mirror switching to 50% for ≤ lμs. Whilst extremely demanding, all of these goals appear to be within reach of current technology. Secondly, initial experimental results are described: (1) the fabrication of a 10μm radius dielectric mirror designed for peak reflectivity R > 0.99997 and the attaching of this to an AFM-type cantilever of mechanical quality Q > 4 x 10⁴ ; (2) the alignment of a cavity of length 2.5cm involving this micro-mirror/cantilever at one end and the demonstration of a finesse of F > 1000 using two independent measurement techniques. The diffraction losses for the cavity are calculated numerically to be < 10^-6 . Other mechanisms limiting the finesse are investigated and the dominant one is determined to be accoustic noise which can be alleviated by placing the cavity into a vacuum. In addition, results demonstrating ultra-fast optical switching of high reflectivity mirrors are shown.

- / -

Η εποχή μας χαρακτηρίζεται από πολλούς ως η ‘’Εποχή της Πληροφορίας’’ και της διακίνησης της. Η καλύτερη λύση σήμερα στα ενσύρματα μέσα μετάδοσης εμφανίζεται να είναι η οπτική ίνα η οποία εξαναγκάζει την τεχνολογία των τηλεπικοινωνιών να μεταπηδήσει από τον κόσμο των ηλεκτρονίων στον κόσμο των φωτονίων. Κατά την μετάδοση της πληροφορίας υπάρχουν όμως φαινόμενα εξασθένησης και παραμόρφωσης του σήματος. Την λύση σε αυτά τα προβλήματα καλείται να δώσει η χρήση σολιτονίων. Στην παρούσα ερευνητική εργασία δημιουργήσαμε έναν αλγόριθμο στην Fortran για να μελετήσουμε την επίδραση διάφορων τύπων διαταραχών στην διάδοση οπτικών σολιτονίων. Συγκεκριμένα πήραμε ένα θεμελιώδες σολιτόνιο αλλά και ανώτερης τάξης και μελετήσαμε την σκέδαση Raman, την αυτό-διαμόρφωση απότομων άκρων (self-steepening) και την διασπορά τρίτης τάξης ξεχωριστά ή και με συνδυασμό αυτών. Επιπλέον χρησιμοποιήσαμε ένα θεμελιώδες σολιτόνιο και για διάφορες τιμές της μιας παραμέτρου το συγκρίναμε με θεωρία. Συμπεράναμε πως η θεωρία διαταραχών προβλέπει σωστά αποτελέσματα με την εξομοίωση για μικρές τιμές των παραμέτρων. / Our era is considered to be the ‘’Era of Information’’ and of its distribution. The optimum solution for the wired communication is considered nowadays to be the fiber optic which forces the telecommunication technology to go over from electrons to photons. During the transmission of information, phenomena as the attenuation and the deformation of the signal, take place. A description of these phenomena is given through the use of Solitons. During this master thesis we built up an algorithm in program language Fortran in order to study the effect of various types of perturbations in optical Solitons. We considered a fundamental and higher-order Solitons and we studied separately the Raman scattering, self-steepening and higher-order dispersion and all together. Also we considered a fundamental Soliton and for various values of one of the three parameters we compared it with the theory expectations. We found out that for small values of the three parameters the perturbation theory provides results which are in alignment with the experiment.

Orientador: Bin Kang Cheng / Dissertação (mestrado) - Universidade Federal do Paraná / Resumo: Neste trabalho de tese estudamos inicialmente o formalismo quântico do potencial com n funções S de Dirac. Em seguida, obtemos os resultados quânticos resolvendo a equação de Schrödinger independente do tempo e usando o método de integral de caminhos. Estudamos dois casos particulares do potencial: (a) potencial não confinado com ri — 2 e (b) potencial confinado com n — 41. Pela solução da equação de Schrödinger independente do tempo podemos determinar completamente a função de onda e os autovalores da energia. Com a integral de caminho obtemos as funções de Green exatas. Para o potencial confinado com n = 4 temos também o traço da função de Green, da qual calculamos os níveis quânticos que estão em total acordo com os resultados obtidos da solução da equação de Schrödinger. / Abstract: In this thesis we first study the quantum formalism of the potential comprising of n Dirac S functions. We present the quantum results both by solving the time independent Schrödinger’s equation and by using the path integral method. We investigate two particular cases of the potential: (a) unbound potential with n = 2 and (b) bound potential with n = 42. From the Schrödinger equation we in fact determine completely the wave function and their energy levels. Applying the path integral method we obtain the exact Green functions and for the bound potential with n = 4 we also obtain the trace of the Green function, from which we evaluate the quantized energies which are in fully agreement with those obtained by solving the Schrödinger’s equation.

Les gaz d'atomes ultra-froids sont des systèmes à N-corps quantiques extrêmement propres et versatiles qui permettent de revisiter dans un environnement contrôlé des concepts fondamentaux souvent issus de la matière condensée. Dans notre système expérimental, nous travaillons avec des gaz d'atomes de potassium 39, qui sont des bosons et qui offrent la possibilité de modifier à loisir les interactions inter-atomiques grâce à des résonances magnétiques de diffusion, ou résonances de Feshbach. Notre équipe s'intéresse particulièrement à la physique des gaz quantiques en basses dimensions et en présence de désordre. Dans un premier temps, nous présentons l’observation et l’étude de la propagation d'un soliton brillant, une onde de matière 1D en interaction attractive, à travers un potentiel désordonné créé à partir d’une figure de speckle optique. Ce travail constitue la première mise en évidence d’effets non-linéaires sur la propagation d’atomes dans un milieu désordonné. Les limites de l’expérience, notamment en terme d’imagerie et de contrôle des champs magnétiques, ont motivé le design et la construction d’une nouvelle enceinte à vide. La suite du manuscrit est dédiée à la description et la caractérisation du nouveau dispositif expérimental, de sa construction à son utilisation pour la production de condensats de Bose-Einstein. La dernière partie de cette thèse est consacrée à l’étude de la localisation d’Anderson d’atomes-froids en présence d’une force constante. La localisation d’Anderson est caractérisée par une suppression de la conductivité sous l’effet du désordre. Dans le cadre des atomes-froids, elle s’explique par la prise en compte de la nature ondulatoire des atomes pendant les processus de diffusion dans le milieu désordonné. Bien qu’à 1D la localisation soit toujours présente, des études théoriques prédisent qu’une force constante appliquée au système modifie de manière drastique les signatures de la localisation (décroissance algébrique de la fonction d’onde) et peut conduire à une délocalisation complète du système. L’étude expérimentale que nous avons menée confirme les prédictions théoriques. / Ultracold atoms gases are quantum many-body systems very clean and versatile which allow to study basic concepts of condensed matter in controlled media. In our experimental system, we work with 39 potassium atoms which are bosons and allow us to modify the interactions between atoms using magnetic resonances of diffusion also called Feshbach resonances. Our team is particulary interested in the physics of quantum gases in low dimension in the presence of disorder. First, we present the observation and study of the propagation of bright solitons, which are 1D matter wave with attractive interactions, through a disordered potentiel made from a speckle pattern of light. This study led to the first observation of nonlinear effects over the propagation of cold atoms in disorder. The limits of the experiment, especially in terms of imaging and magnetic field control, motivated the design and construction of a new vacuum chamber. The next part of the manuscript is dedicated to the description and characterization of the new experimental device, from its construction to its use for the production of Bose-Einstein condensates. The last part of this thesis is devoted to the study of Anderson localization of cold atoms in the presence of a constant bias force. Anderson localization is characterized by a suppression of conductivity under the effect of disorder. In the context of the cold atoms, it is explained by taking into account the wave nature of the atoms during the diffusion processes in the disordered medium. Even if localization is always present in 1D systems, theoretical studies predict that a constant force applied to the system drastically modifies the signatures of the localization (algebraic decay of the wave function) and can lead to a complete delocalization of the system. Our experimental study confirms the theoretical predictions.

Two separate problems are to be investigated in this thesis. The first problem is the propagation of waves through a short rod (or slug) of viscoelastic material. The second problem is the study of impact of a short viscoelastic rod (or slug) on a stationary semi-infinite viscoelastic. rod. The viscoelastic materials are modelled as standard linear solids which involve 3 material parameters and the motion is treated. as one-dimensional. For the first study, a viscoelastic slug is placed between two semi-infinite elastic rods and a wave initiated in the first rod is transmitted through the slug into the second rod. The objective is to relate the transmitted signal to the material parameters of the slug. We solve the governing system of partial differential equations using the Laplace transform and we examine the propagating velocity discontinuity using discontinuity analysis and the limit theorem of the Laplace transform. We then approximate the solution of the propagating disturbance using the regular perturbation method. We invert the Laplace transformed solution numerically to obtain the transmitted signal for several viscosity time constants and ratios of acoustic impedances. We compare the results obtained using the above techniques. In the second problem, we first model the impact and solve the governing system of partial differential equations in the Laplace transform domain. Then we examine the propagating stress and velocity discontinuities using discontinuity analysis. We approximate the solutions of the propagating stress and velocity using the regular and multiple scales perturbation methods. In this problem, we first consider the slug is elastic and the rod is viscoelastic. Secondly, we consider the slug is viscoelastic and the rod is elastic and thirdly, we consider both materials are viscoelastic. Numerically we invert the Laplace transformed solutions for the interface stress and interface velocity for several viscosity time constants and ratios of acoustic impedances to determine whether the slug and the rod part company or remain in contact. Then we compare the results obtained using the discontinuity analysis, regular and multiple scales perturbation methods.

Dans cette thèse, nous prouvons des résultats concernant le contrôle et la stabilisation d'équations dispersives étudiées sur un intervalle borné. Pour commencer, nous étudions la stabilisation interne du système de Gear-Grimshaw, qui est un système de deux équations de Korteweg-de-Vries (KdV) couplées. Nous obtenons une décroissance exponentielle de l'énergie totale associée au modèle en introduisant une fonction de Lyapunov convenable. Nous prouvons aussi des résultats de contrôlabilité à zéro et exacte pour l'équation de Korteweg-de Vries avec un contrôle distribué à support dans un sous-intervalle du domaine. Pour la contrôlabilité à zéro du système linéarisé, nous utilisons l'approche classique basée sur la dualité qui ramène le problème à l'étude d'une inégalité d'observabilité qui, dans ce travail, est établie à l'aide d'une inégalité de Carleman. Ensuite, utilisant des fonctions plateau, nous prouvons un résultat de contrôlabilité exacte. Dans les deux cas, le résultat concernant le système non linéaire est obtenu à l'aide d'un argument de point fixe. Enfin, dans la lignée du résultat de contrôlabilité au bord obtenu par L. Rosier pour KdV, nous prouvons que le système linéaire de Boussinesq de type KdV-KdV est exactement contrôlable lorsque des contrôles sont appliqués au bord. Notre méthode repose sur l'utilisation de multiplicateurs et l'approche de la dualité mentionnée ci-dessus. Lorsqu'un mécanisme d'amortissement est introduit au bord, nous montrons que le système non linéaire est aussi exactement contrôlable et que l'énergie associée au modèle décroit exponentiellement / This work is devoted to prove a series of results concerning the control and stabilization properties of dispersive models posed on a bounded interval. Initially, we study the internal stabilization of a coupled system of two Korteweg-de Vries equations (KdV), the so-called Gear-Grimshaw system. Defining a convenient Lyapunov function we obtain the exponential decay of the total energy associated to the model. We also prove results of null and exact controllability for the Korteweg-de Vries equation with a control acting internally on a subset of the domain. In the case of the null controllability for the linear model, we use a classical duality approach which reduces the problem to the study of an observability inequality that, in this work, is proved by means of a Carleman inequality. Then, making use of cut-off functions, the exact controllability is also investigated. In both cases, the result for the nonlinear system is obtained by means of fixed-point argument. Finally, in view of the result of the boundary controllability obtained by L. Rosier for the KdV equation, we prove that the linear Boussinesq system of KdV-KdV type is exactly controllable when the controls act in the boundary conditions. Our analysis is performed using multipliers and the duality approach mentioned above. Adding a damping mechanism in the boundary, it is proved that the nonlinear system is also exactly controllable and that the energy associated to the model decays exponentially