Universiẗat, Diss., 2002--Jena.

Techn. University, Diss., 2004--Clausthal.

University, Diss., 2004--Osnabrück.

This thesis is missing pages xiii, xiv and xv. None of which are in other copies of the thesis. -Digitization Centre / Solitons and soliton systems have introduced many interests in the applications of signal processing and communication systems due to their special properties. To facilitate the various applications, a digital soliton system is designed to overcome the inherent drawbacks of traditional analog soliton systems in this thesis. Wave digital theory is employed to design a digital model of the nonlinear Toda lattice circuit. The designed model is implemented in Simulink, and numerical results of the simulation verifies the important properties of the digital model and show it to be a good digital soliton system simulator. Moreover, an example of a soliton communication system is provided to demonstrate the digital soliton system simulator can work as well in soliton communication systems, avoiding the inherent problems of analog implementations. In addition, the digital Toda lattice circuit modelled in Simulink can be customized to run in DSP and FPGA. Such hardware co-processing will highly improve the speed of the simulation processes. / Thesis / Master of Applied Science (MASc)

Assumindo que a matéria nuclear seja um fluido perfeito, estudamos a propagação de perturbações na densidade bariônica. A equação de estado é obtida através de um modelo relativístico em campo médio, o qual é uma variante do modelo não-linear de Walecka. A expansão das equações de Euler e da continuidade na hidrodinâmica relativística em torno das configurações de equilíbrio nos levam a equações diferenciais para a perturbação na densidade. Resolvemos tais equações numericamente para perturbações lineares e esféricas mediante pulsos iniciais. Para perturbações lineares econtramos soluções solitônicas de pulsos isolados e soluções com vários solitons seguidas de ``radiação\'\'. Dependendo da equação de estado um forte amortecimento pode ocorrer. Consideramos também a evolução de perturbações em um meio sem efeitos dissipativos. Nesse caso observamos a formação e quebra de ondas de choque. Depois estudamos todo o formalismo na matéria nuclear em temperatura finita. Nossos resultados podem ser relevantes para análise de dados do RHIC. Eles sugerem que ondas de choque formadas na fase de plasma de quarks e gluons podem sobreviver e se propagar na fase hadrônica. Também estudamos a equação de onda não-linear para perturbações na densidade bariônica e densidade de energia no plasma de quarks e gluons (QGP). Sob certas condições solitons podem existir no QGP. Finalmente discutimos métodos alternativos de soluções de equações di-ferenciais não-lineares. / Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density. The equation of state is derived from a relativistic mean field model, which is a variant of the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations leads to differential equations for the density perturbation. We solve them numerically for linear and spherical perturbations and follow the propagation of the initial pulses. For linear perturbations we find single soliton solutions and solutions with one or more solitons followed by ``radiation\'\'. Depending on the equation of state a strong damping may occur. We consider also the evolution of perturbations in a medium without dispersive effects. In this case we observe the formation and breaking of shock waves. We study all these equations also for matter at finite temperature. Our results may be relevant for the analysis of RHIC data. They suggest that the shock waves formed in the quark gluon plasma phase may survive and propagate in the hadronic phase. We also study the non-linear wave equation for pertubations in baryon density and energy density in quark-gluon-plasma (QGP). Under certains conditions solitons may exist in QGP. Finally we discuss alternatives methods for solving non-linear differential equations.

The main subject of this thesis is solitons due to degenerate parametric four-wave mixing. Derivation of the governing equations is carried out for both spatial solitons (slab waveguide) and temporal solitons (optical fibre). Higher-order effects that are ignored in the standard paraxial approximation are discussed and estimated. Detailed analysis of conventional solitons is carried out. This includes discovery of various solitons families, linear stability analysis of fundamental and higher-order solitons, development of theory describing nonlinear dynamics of higher-order solitons. The major findings related to the stationary problem are bifurcation of a two-frequency soliton family from an asymptotic family of infinitely separated one-frequency solitons, jump bifurcation and violation of the bound state principle. Linear stability analysis shows a rich variety of internal modes of the fundamental solitons and existence of a stability window for higher-order solitons. Theory for nonlinear dynamics of higher-order solitons successfully predicts the position and size of the stability window, and various instability scenarios. Equivalence between direct asymptotic approach and invariant based approach is demonstrated. A general analytic approach for description of localised solutions that are in resonance with linear waves (quasi-solitons and embedded solitons) is given. This includes normal form theory and approximation of interacting particles. The main results are an expression for the amplitude of the radiating tail of a quasi-soliton, and a two-fold criterion for existence of embedded solitons. Influence of nonparaxiality on soliton stability is investigated. Stationary instability threshold is derived. The major results are shift and decreasing of the size of the stability window for higher-order solitons. The latter is the first demonstration of the destabilizing influence of nonparaxiality on higher-order solitons. Analysis of different aspects of solitons is based on universal approaches and methods. This includes Hamiltonian formalism, consideration of symmetry properties of the model, development of asymptotic models, construction of perturbation theory, application of general theorems etc. Thus, the results obtained can be extended beyond the particular model of degenerate four-wave mixing. All theoretical predictions are in good agreement with the results of direct numerical modeling.

Assumindo que a matéria nuclear seja um fluido perfeito, estudamos a propagação de perturbações na densidade bariônica. A equação de estado é obtida através de um modelo relativístico em campo médio, o qual é uma variante do modelo não-linear de Walecka. A expansão das equações de Euler e da continuidade na hidrodinâmica relativística em torno das configurações de equilíbrio nos levam a equações diferenciais para a perturbação na densidade. Resolvemos tais equações numericamente para perturbações lineares e esféricas mediante pulsos iniciais. Para perturbações lineares econtramos soluções solitônicas de pulsos isolados e soluções com vários solitons seguidas de ``radiação\'\'. Dependendo da equação de estado um forte amortecimento pode ocorrer. Consideramos também a evolução de perturbações em um meio sem efeitos dissipativos. Nesse caso observamos a formação e quebra de ondas de choque. Depois estudamos todo o formalismo na matéria nuclear em temperatura finita. Nossos resultados podem ser relevantes para análise de dados do RHIC. Eles sugerem que ondas de choque formadas na fase de plasma de quarks e gluons podem sobreviver e se propagar na fase hadrônica. Também estudamos a equação de onda não-linear para perturbações na densidade bariônica e densidade de energia no plasma de quarks e gluons (QGP). Sob certas condições solitons podem existir no QGP. Finalmente discutimos métodos alternativos de soluções de equações di-ferenciais não-lineares. / Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density. The equation of state is derived from a relativistic mean field model, which is a variant of the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations leads to differential equations for the density perturbation. We solve them numerically for linear and spherical perturbations and follow the propagation of the initial pulses. For linear perturbations we find single soliton solutions and solutions with one or more solitons followed by ``radiation\'\'. Depending on the equation of state a strong damping may occur. We consider also the evolution of perturbations in a medium without dispersive effects. In this case we observe the formation and breaking of shock waves. We study all these equations also for matter at finite temperature. Our results may be relevant for the analysis of RHIC data. They suggest that the shock waves formed in the quark gluon plasma phase may survive and propagate in the hadronic phase. We also study the non-linear wave equation for pertubations in baryon density and energy density in quark-gluon-plasma (QGP). Under certains conditions solitons may exist in QGP. Finally we discuss alternatives methods for solving non-linear differential equations.

C'est autour de la photoréfractivité et de l'auto-focalisation laser que mes travaux de recherche<br />ont principalement été effectués. Cet ouvrage tente une description qui se veut la plus complète<br />possible sans toutefois sacrifier à la minutie des détails, laquelle se trouvant dans d'autres écrits<br />plus spécialisés tels que thèses ou publications scientifiques.<br />Nous la démarrons par une introduction aux raisons qui nous ont conduit à mener ces travaux.<br />Les premiers développements de l'auto-focalisation laser photoréfractive sont donnés dans<br />la première partie — laquelle contient également des références à des travaux antérieurs et à<br />des activités d'enseignement. La deuxième est consacrée aux approfondissements qui ont suivi et<br />qui, pour la plupart, sont encore en cours. La troisième partie est, elle, dévolue à des sujets en<br />marge, qui ont en commun avec les premiers d'une part la photoréfractivité et d'autre part l'autofocalisation.<br />Par ailleurs, c'est en annexe que l'on trouvera des détails mathématiques originaux<br />ainsi que des listes de publications dont certaines sont données dans leur totalité. La bibliographie<br />s'y trouve également à sa place, la dernière.<br />Nous nous attachons à démontrer expérimentalement la réalité de l'auto-focalisation d'un faisceau<br />laser dans un matériau photoréfractif, à des temps aussi de l'ordre de la seconde que de<br />la nanoseconde. Nous avons également développé deux modèles distincts qui nous permettent<br />d'interpréter nos résultats. L'exploitation de ces modèles passe par la réalisation d'algorithmes parall`<br />eles, au développement desquels quelques pages sont consacrées. Ces études ont permis, de par<br />les outils qu'elles utilisent, le développement d'autres thématiques —Double Conjugaison de Phase<br />Photoréfractive et Auto-focalisation dans les solutions photopolymérisables—, au sujet desquelles<br />la dernière partie de ce document apporte quelques éclaircissements.

Investigating the behavior of an atom in response to the applied electric field when the latter frequency is in resonance with the natural frequency of the atom is an interesting subject. Near resonance, some of the most interesting optical phenomena such as dispersion, and absorption are more pronounced. Moreover, considering resonant and near resonant interaction of light with two-level atoms, uncovers fascinating physical phenomena such as area theorem and self-induced transparency describing stable pulse propagation regimes in which the pulses maintain their identity. Existence of these optical structures in on-resonance optical media has not been investigated in detail. In this thesis, the on-resonance interaction of optical field with atoms is considered and the formation of several novel self-similar and kink waves in linear and nonlinear resonant media is discovered and theoretically explored. First, self-similar pulse formation in homogeneous broadened linear amplifiers in a vicinity of an optical resonance is analyzed. It is demonstrated that the self-similar pulses serve as universal asymptotics of any near resonance short pulses propagating in coherent linear amplifiers. Second, ultrashort self-similar pulse propagation in coherent linear absorbers near optical resonance is investigated. Third, existence of self-similar optical waves with kink structure in resonant optical systems is discovered. Fourth, it is found that self-induced transparency quadratic solitons are realizable in the media with quadratic optical nonlinearities, doped with resonant impurities. Finally, stable spatial similaritons supported by homogeneous conservative optical media with quintic nonlinearities are explored. To experimentally realize the presented results, physical models are presented for all systems under consideration. The stability of the proposed near resonance optical systems is demonstrated through a series of numerical case studies.

The structure of protein is essentially important for life activities. Proteins can perform their functions only by specific structures. In this thesis, the kink and multi-kink model for protein description are reviewed. It is shown that most of the loop parts in Protein Databank (PDB) can be described by very limited number of kinks within the experimental precision. Furthermore, by applying the model into two well studied real proteins (myoglobin and villin headpiece HP35), it is shown that the multi-kink model gives correct folding pathway and thermal dynamical properties compared with the experimental results for both proteins. In particular, the kink model is computationally inexpensive compared with other existing models. In the last chapter, a new visualization method for the heavy atoms in the side-chain is presented.