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End date: 2021 × End date: 2019 × Start date: 2019 × Has files: Yes × Type: Report ×

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Now showing 1 - 6 of 6
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    Temporal dissipative solitons in time-delay feedback systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Yanchuk, Serhiy; Ruschel, Stefan; Sieber, Jan; Wolfrum, Matthias
    Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh-Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability.
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    Inverse elastic scattering from rigid scatterers with a single incoming wave : this paper is dedicated to the memory of Armin Lechleiter
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Elschner, Johannes; Hu, Guanghui
    The @first part of this paper is concerned with uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.
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    Generalized gradients for probabilistic/robust (probust) constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Ackooij, Wim van; Henrion, René; Pérez-Aros, Pedro
    Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data.
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    Exponential moments for planar tessellations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Tóbiás, András; Jahnel, Benedikt
    In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.
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    Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Emmrich, Etienne; Lasarzik, Robert
    A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.
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    A large-deviations approach to gelation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Andreis, Luisa; König, Wolfgang; Patterson, Robert
    A @large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t = 1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdos-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdos-Rényi graphs are connected.