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Wagner W.,Weierstrass Institute for Applied Analysis And Stochastics
Physics of Fluids | Year: 2011

The paper is concerned with some aspects of stochastic modeling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (monomolecular processes) or via binary interactions (bimolecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent. © 2011 American Institute of Physics. Source

Driben R.,Tel Aviv University | Babushkin I.,Weierstrass Institute for Applied Analysis And Stochastics
Optics Letters | Year: 2012

Soliton fusion is a fascinating and delicate phenomenon that manifests itself in optical fibers in case of interaction between copropagating solitons with small temporal and wavelength separation. We show that the mechanism of acceleration of a trailing soliton by dispersive waves radiated from the preceding one provides necessary conditions for soliton fusion at the advanced stage of supercontinuum generation in photonic-crystal fibers. As a result of fusion, large-intensity robust light structures arise and propagate over significant distances. In the presence of small random noise the delicate condition for the effective fusion between solitons can easily be broken, making the fusion-induced giant waves a rare statistical event. Thus oblong-shaped giant accelerated waves become excellent candidates for optical rogue waves. © 2012 Optical Society of America. Source

Purwins H.-G.,Institute For Angewandte Physik | Bodeker H.U.,Institute For Angewandte Physik | Amiranashvili S.,Institute For Angewandte Physik | Amiranashvili S.,Weierstrass Institute for Applied Analysis And Stochastics
Advances in Physics | Year: 2010

The present review summarizes experimental and theoretical work dealing with self-organized solitary localized structures (LSs) that are observed in spatially extended nonlinear dissipative systems otherwise exhibiting translational and rotational symmetry. Thereby we focus on those LSs that essentially behave like particles and that we call dissipative solitons (DSs). Such objects are also solutions of corresponding nonlinear evolution equations and it turns out that they are rather robust with respect to interaction with each other, with impurities, and with the boundary; alternatively they are generated or annihilated as a whole. By reviewing the experimental results it turns out that the richest variety of DS phenomena has been observed in electrical transport systems and optical devices. Nevertheless, DSs show up also in many other systems, among which nerve pulses in living beings are of uppermost importance in practice. In most of these systems DSs behave very similarly. The experimental results strongly suggest that phenomenon of DSs is universal. On the background of the experimental findings models for a theoretical understanding are discussed. It turns out that in a limited number of cases a straightforward quantitative description of DS patterns can be carried out. However, for the overwhelming number of systems only a qualitative approach has been successful so far. In the present review particular emphasis is laid on reaction-diffusion systems for which a kind of 'normal form' can be written down that defines a relatively large universality class comprising e.g. important electrical transport, chemical, and biological systems. For the other large class of DS carrying systems, namely optical devices, the variety of model equations is much larger and one is far away, even from a universal qualitative description. Because of this, and due to the existence of several extensive reviews on optical systems, their theoretical treatment has been mentioned only shortly. Finally, it is demonstrated that in terms of a singular perturbation approach the interaction of DSs and important aspects of their bifurcation behaviour, under certain conditions, can be described by rather simple equations. This is also true when deriving from the underlying field equations a set of ordinary differential equations containing the position coordinates of the individual DSs. Such equations represent a theoretical foundation of the experimentally observed particle-like behaviour of DSs. Though at present there is little real practical application of DSs and related patterns in an outlook we point out in which respects this might change in future. A systematic summary of a large amount of experimental and theoretical results on reaction-diffusion systems, being rather close to the subject of the present review, can also be found on the website http://www.uni-muenster.de/Physik.AP/Purwins/Research- Summary. © 2010 Taylor & Francis. Source

Wolfrum M.,Weierstrass Institute for Applied Analysis And Stochastics
Physica D: Nonlinear Phenomena | Year: 2012

We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimension-two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes. © 2012 Elsevier B.V. All rights reserved. Source

Si H.,Weierstrass Institute for Applied Analysis And Stochastics
Finite Elements in Analysis and Design | Year: 2010

A constrained Delaunay tetrahedralization of a domain in R3 is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is on the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions. © 2009 Elsevier B.V. All rights reserved. Source

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