2011, Amiranashvili, Sh., Demircan, A.

We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrdinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse.

2010, Münch, Andreas, Wagner, Barbara

In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into ``fingers''. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.

2010, Haller-Dintelmann, Robert, Höppner, Wolfgang, Kaiser, Hans-Christoph, Rehberg, Joachim, Ziegler, Günter M.

We study relative stability properties of different clusters of closely packed one- and two-dimensional localized peaks of the Swift-Hohenberg equation. We demonstrate the existence of a 'spatial Maxwell' point where clusters are almost equally stable, irrespective of the number of pes involved. Above (below) the Maxwell point, clusters become more (less) stable with the increase of the number of peaks

2011, Paoli, Laetitia, Petrov, Adrien, Schatzman, M.

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.

2011, Tabelow, K., Polzehl, J.

The purpose of the package fmri is the analysis of single subject functional magnetic resonance imaging (fMRI) data. It provides fMRI analysis from time series modeling by a linear model to signal detection and publication quality images. Specifically, it implements structural adaptive smoothing methods with signal detection for adaptive noise reduction which avoids blurring of activation areas. Within this paper we describe the complete pipeline for fMRI analysis using fmri. We describe data reading from various medical imaging formats and the linear modeling used to create the statistical parametric maps. We review the rationale behind the structural adaptive smoothing algorithms and explain their usage from the package fmri. We demonstrate the results of such analysis using two experimental datasets. Finally, we report on the usage of a graphical user interface for some of the package functions.

2010, Bartels, Sören, Müller, Rüdiger

A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.

2011, Paoli, Laetitia, Petrov, Adrien

This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.

2012, Otto, C., Lüdge, K., Vladimirov, A.G., Wolfrum, M., Schöll, E.

We study a passively mode-locked semiconductor ring laser subject to optical feedback from an external mirror. Using a delay differential equation model for the mode-locked laser, we are able to systematically investigate the resonance effects of the inter-spike interval time of the laser and the roundtrip time of the light in the external cavity (delay time) for intermediate and long delay times. We observe synchronization plateaus following the ordering of the well-known Farey sequence. Our results show that in agreement with the experimental results a reduction of the timing jitter is possible if the delay time is chosen close to an integer multiple of the inter-spike interval time of the laser without external feedback. Outside the main resonant regimes the timing jitter is drastically increased by the feedback.

2012, Gün, Onur, König, Wolfgang, Sekulov´c, Ozren

We consider the long-time behaviour of a branching random walk in random environment on the lattice Zd. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments m_np , i.e., the p-th moments over the medium of the n-th moment over the migration and killing/branching, of the local and global population sizes. For n = 1, this is well-understood citeGM98, as m_1 is closely connected with the parabolic Anderson model. For some special distributions, citeA00 extended this to ngeq2, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for m_n. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that m_n^p m_1^np are asymptotically equal, up to an error e^o(t). The cornerstone of our method is a direct Feynman-Kac-type formula for mn, which we establish using the spine techniques developed in citeHR1.1

2011, Paoli, Laetitia, Petrov, Adrien, Gröger, K.

We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.