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Subject of this work is the investigation of universal scaling laws which are observed in coupled chaotic systems. Progress is made by replacing the chaotic fluctuations in the perturbation dynamics by stochastic processes. First, a continuous-time stochastic model for weakly coupled chaotic systems is introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck equation scaling relations are derived, which are confirmed by results of numerical simulations. Next, the new effect of avoided crossing of Lyapunov exponents of weakly coupled disordered chaotic systems is described, which is qualitatively similar to the energy level repulsion in quantum systems. Using the scaling relations obtained for the coupling sensitivity of chaos, an asymptotic expression for the distribution function of small spacings between Lyapunov exponents is derived and compared with results of numerical simulations. Finally, the synchronization transition in strongly coupled spatially extended chaotic systems is shown to resemble a continuous phase transition, with the coupling strength and the synchronization error as control and order parameter, respectively. Using results of numerical simulations and theoretical considerations in terms of a multiplicative noise partial differential equation, the universality classes of the observed two types of transition are determined (Kardar-Parisi-Zhang equation with saturating term, directed percolation).

In the present work, we study wave phenomena in strongly nonlinear lattices. Such lattices are characterized by the absence of classical linear waves. We demonstrate that compactons – strongly localized solitary waves with tails decaying faster than exponential – exist and that they play a major role in the dynamics of the system under consideration. We investigate compactons in different physical setups. One part deals with lattices of dispersively coupled limit cycle oscillators which find various applications in natural sciences such as Josephson junction arrays or coupled Ginzburg-Landau equations. Another part deals with Hamiltonian lattices. Here, a prominent example in which compactons can be found is the granular chain. In the third part, we study systems which are related to the discrete nonlinear Schrödinger equation describing, for example, coupled optical wave-guides or the dynamics of Bose-Einstein condensates in optical lattices. Our investigations are based on a numerical method to solve the traveling wave equation. This results in a quasi-exact solution (up to numerical errors) which is the compacton. Another ansatz which is employed throughout this work is the quasi-continuous approximation where the lattice is described by a continuous medium. Here, compactons are found analytically, but they are defined on a truly compact support. Remarkably, both ways give similar qualitative and quantitative results. Additionally, we study the dynamical properties of compactons by means of numerical simulation of the lattice equations. Especially, we concentrate on their emergence from physically realizable initial conditions as well as on their stability due to collisions. We show that the collisions are not exactly elastic but that a small part of the energy remains at the location of the collision. In finite lattices, this remaining part will then trigger a multiple scattering process resulting in a chaotic state.

Since their discovery in 1610 by Galileo Galilei, Saturn's rings continue to fascinate both experts and amateurs. Countless numbers of icy grains in almost Keplerian orbits reveal a wealth of structures such as ringlets, voids and gaps, wakes and waves, and many more. Grains are found to increase in size with increasing radial distance to Saturn. Recently discovered "propeller" structures in the Cassini spacecraft data, provide evidence for the existence of embedded moonlets. In the wake of these findings, the discussion resumes about origin and evolution of planetary rings, and growth processes in tidal environments. In this thesis, a contact model for binary adhesive, viscoelastic collisions is developed that accounts for agglomeration as well as restitution. Collisional outcomes are crucially determined by the impact speed and masses of the collision partners and yield a maximal impact velocity at which agglomeration still occurs. Based on the latter, a self-consistent kinetic concept is proposed. The model considers all possible collisional outcomes as there are coagulation, restitution, and fragmentation. Emphasizing the evolution of the mass spectrum and furthermore concentrating on coagulation alone, a coagulation equation, including a restricted sticking probability is derived. The otherwise phenomenological Smoluchowski equation is reproduced from basic principles and denotes a limit case to the derived coagulation equation. Qualitative and quantitative analysis of the relevance of adhesion to force-free granular gases and to those under the influence of Keplerian shear is investigated. Capture probability, agglomerate stability, and the mass spectrum evolution are investigated in the context of adhesive interactions. A size dependent radial limit distance from the central planet is obtained refining the Roche criterion. Furthermore, capture probability in the presence of adhesion is generally different compared to the case of pure gravitational capture. In contrast to a Smoluchowski-type evolution of the mass spectrum, numerical simulations of the obtained coagulation equation revealed, that a transition from smaller grains to larger bodies cannot occur via a collisional cascade alone. For parameters used in this study, effective growth ceases at an average size of centimeters.

The theory of atomic Boson-Fermion mixtures in the dilute limit beyond mean-field is considered in this thesis. Extending the formalism of quantum field theory we derived expressions for the quasi-particle excitation spectra, the ground state energy, and related quantities for a homogenous system to first order in the dilute gas parameter. In the framework of density functional theory we could carry over the previous results to inhomogeneous systems. We then determined to density distributions for various parameter values and identified three different phase regions: (i) a stable mixed regime, (ii) a phase separated regime, and (iii) a collapsed regime. We found a significant contribution of exchange-correlation effects in the latter case. Next, we determined the shift of the Bose-Einstein condensation temperature caused by Boson-Fermion interactions in a harmonic trap due to redistribution of the density profiles. We then considered Boson-Fermion mixtures in optical lattices. We calculated the criterion for stability against phase separation, identified the Mott-insulating and superfluid regimes both, analytically within a mean-field calculation, and numerically by virtue of a Gutzwiller Ansatz. We also found new frustrated ground states in the limit of very strong lattices. ----Anmerkung: Der Autor ist Träger des durch die Physikalische Gesellschaft zu Berlin vergebenen Carl-Ramsauer-Preises 2004 für die jeweils beste Dissertation der vier Universitäten Freie Universität Berlin, Humboldt-Universität zu Berlin, Technische Universität Berlin und Universität Potsdam.