Browsing by Author "Schneider, Guido (Prof. Dr.)"

Now showing 1 - 14 of 14

Open Access
Amplitude equations for Boussinesq and Ginzburg-Landau-like models
(2019) Haas, Tobias; Schneider, Guido (Prof. Dr.)
Open Access
Effective equations in mathematical quantum mechanics
(2017) Gilg, Steffen; Schneider, Guido (Prof. Dr.)
Open Access
Existence and non-existence of breather solutions on necklace graphs
(2023) Kielwein, Tobias; Schneider, Guido (Prof. Dr.)
Open Access
Failure of amplitude equations
(2016) Sunny, Danish Ali; Schneider, Guido (Prof. Dr.)
The nonlinear Schrodinger (NLS) equation is an example of a universal nonlinear model that describes many physical nonlinear systems. The equation can be applied to hydrodynamics, nonlinear optics, nonlinear acoustics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. It describes small modulations in time and space of a spatially and temporally oscillating wave packet advancing in a laboratory frame. It has first been derived for the so called water wave problem in 1968 and the proof that it makes correct predictions has been recently the subject of intensive research.
Open Access
Invasion phenomena in pattern-forming systems admitting a conservation law structure
(2021) Hilder, Bastian; Schneider, Guido (Prof. Dr.)
Open Access
Justification of an approximation equation for the Benard-Marangoni problem
(2014) Zimmermann, Dominik; Schneider, Guido (Prof. Dr.)
The Benard-Marangoni problem is a mathematical model for the description of a temperature dependent fluid flow in very thin liquid layers with a free top surface. The liquid is bounded from below by a horizontal plate of a certain temperature. Above the liquid there is an atmosphere cooler than the bottom plate. There is a purely conducting steady state, where the liquid is at rest. This state is stable as long as the difference between the temperature of the bottom plate and the temperature of the atmosphere is sufficiently small. If the temperature difference surpasses a certain threshold, convection sets in, which is mainly driven by surface tension rather than buoyancy. The onset of convection can be seen as the propagation of a spatially periodic pattern, such that we interpret the Benard-Marangoni problem as a pattern forming system. In this thesis we are interested in the behaviour of the system when the purely conducting steady state becomes unstable. From the equations of the Benard-Marangoni problem we formally derive a Ginzburg-Landau like system of modulation equations, which we use to construct approximate solutions for the full problem. In this thesis we prove an approximation theorem for these modulation equations. That means, we show that the approximate solutions lie close to true solutions of the Benard-Marangoni problem, at least for a long time. The validity of the Ginzburg-Landau approximation was already shown for a number of pattern forming systems. In case of the Benard-Marangoni problem, however, we have a spectral situation that does not allow a direct application of the existing approximation proofs. Hence, we first consider a toy problem exhibiting such a kind of spectrum and develop a method for proving an approximation result in this case. Furthermore, the existing approximation proofs were restricted to semilinear problems. However, the equations of the Benard-Marangoni problem are quasilinear. Therefore, we also develop a method for proving approximation results for quasilinear problems. We then turn back to the Benard-Marangoni problem. After showing local existence and uniqueness of solutions, we apply our new methods in order to prove the desired approximation result.
Open Access
The KdV and Whitham limit for a spatially periodic Boussinesq model
(2016) Bauer, Roman; Schneider, Guido (Prof. Dr.)
Wir betrachten die KdV-Approximation und die Whitham-Approximation für ein räumlich periodisches Boussinesq-Modell. Wir zeigen Abschätzungen der Differenz zwischen der KdV- beziehungsweise der Whitham-Approximation und echten Lösungen des ursprünglichen Modells, welche garantieren, dass diese Amplituden-Gleichungen korrekte Vorhersagen über die Dynamik des räumlich periodischen Boussinesq-Modells über die natürlichen Zeitskalen machen. Der Beweis basiert auf Blochwellenanalysis und Energieabschätzungen.
Open Access
Long wave approximation over and beyond the natural time scale
(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)
Open Access
Mathematical modeling and numerical simulations of the extrinsic pro-apoptotic signaling pathway
(2013) Daub, Markus; Schneider, Guido (Prof. Dr.)
Apoptosis is a special form of programmed cell death and plays an important role in cancer research, for instance. An essential part of the apoptotic process is the extrinsic signaling pathway, which is initiated at the cellular membrane by active ligand-receptor clusters. The activation of special proteins, so-called caspases, starts a caspase cascade, a reaction network of diverse caspase types, that finally results in the death of the cell. The extrinsic signaling pathway can simply be separated into two components: on the one hand, the formation of ligand-receptor clusters on the cellular membrane which act as stimulus of the signaling pathway, and on the other hand, the intracellular reaction network where diverse caspase types are involved in. In this thesis, mathematical models for these two processes are introduced and analyzed. According to the structure of the signaling pathway, the thesis is separated into two parts: 1) In the first part of the thesis, we introduce a spatially extended reaction-diffusion model for the reaction network with diverse caspase types. First, we show that the system has one unstable and two asymptotically stable, spatially homogeneous, stationary solutions. Moreover, we prove that the reaction-diffusion system admits a bounded invariant region that guarantees the boundedness of the solution for all time. After the analytic part of the first chapter, we proceed with a numerical analysis of the reaction-diffusion system. The numerical simulation shows that the diffusion rapidly balances the caspase concentration and that the behavior of the system is subsequently described by the reaction kinetics. We use this fact for the investigation of the long-time behavior of the reaction-diffusion system and split the simulation into two steps. With numerical simulations, we show that the system is switch-like concerning the initial condition with the outputs "cell death" or "cell survival". A typical structure of solutions of bistable reaction-diffusion systems is a traveling wave. Solving the initial value problem numerically presents traveling wave solutions, and we determine their velocity and direction of travel. However, traveling wave solutions only exist on a large spatial scale. 2) In the second part of the thesis, we introduce a particle model for the motion of the receptors and ligands on the cell membrane in order to investigate the formation of ligand-receptor clusters. The motion of the particles is random and modeled as a Brownian motion. Additionally, we take into account the interaction between the ligands and receptors and derive stochastic differential equations for the translation and rotation of the particles. Since we consider a huge amount of receptors and ligands, we obtain a high-dimensional system of stochastic differential equations which are nonlinearly coupled by the mutual interaction of the particles. The flexible form of the particle model enables the comparison of different biological hypotheses concerning the binding behavior of the receptors. Due to the high complexity of the simulation, the algorithms implemented in the computer language C are mapped to GPGPU architectures in order to parallelize the computation of the interactions. This enables us to perform simulations for various particle configurations and compare the results concerning the formation of signal competent cluster units. In order to analyze large ligand-receptor clusters and especially their structure, we apply the visualization-tool cellVis.
Open Access
Mathematische Modellierung von wellenoptischer Absorption beim Laserschneiden
(2022) Wörner, Maximilian; Schneider, Guido (Prof. Dr.)
Open Access
Nonlinear phenomena on metric and discrete necklace graphs
(2019) Maier, Daniela; Schneider, Guido (Prof. Dr.)
Open Access
Singular limits in KGZ systems and the DNLS approximation in case of quadratic nonlinearities
(2024) Taraca, Raphael; Schneider, Guido (Prof. Dr.)
Open Access
Validity and attractivity of amplitude equations
(2016) Sanei Kashani, Kourosh; Schneider, Guido (Prof. Dr.)
Whitham's equations and the Ginzburg-Landau equation belong to a set of famous amplitude equations containing the KdV equation, the NLS equation, Burgers equation, and so-called phase diffusion equations. They play an important role in the description of spatially extended dissipative or conservative physical systems. Except of Whitham's system for all other amplitude equations there exists a satisfying mathematical theory showing that the original system behaves approximately as predicted by the associated amplitude equation. In the first part of this work we therefore derive Whitham's equations for a coupled system of equations, namely a Klein-Gordon-Boussinesq model. Subsequently we prove the validity of Whitham's equations for this system. The combination of our scaled ansatz adapted to Whitham's equations with the resonance structure of our system poses a new challenge. In order to prove the approximation results for Whitham's equations we will require some infinite series of normal transformations, for which we need to prove the convergence. In the second part we prove the attractivity of the Ginzburg-Landau manifold for a toy problem inspired by Marangoni convection. In comparison to the previous classical situation in our case the curve of eigenvalues possesses additionally a marginally stable mode at the origin. Therefore, we will need to modify the requirements for the attractivity result and the method of proof.
Open Access
The validity of the Nonlinear Schrödinger approximation in higher space dimensions
(2014) Hermann, Alina; Schneider, Guido (Prof. Dr.)
The goal of the present work is the proof of approximation results for the Nonlinear Schördinger approximation in higher space dimensions for dispersive systems. The focus is on systems with resonant quadratic terms, which can lead to some explosion before the end of the approximation interval. In higher space dimensions the resonance structure is much more complicated than in case of one space dimension. The proof of approximation results is based on normal form transforms and the use of time-dependent norms.