Rademacher, Jens

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Rademacher, Jens
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Rademacher, Jens
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    Item-typ:Veröffentlichung,
    Diffusion, Advection and Pattern Formation
    (2020-03-23)
    Yang, Jichen
    ;
    Kühn, Christian
    ;
    Rademacher, Jens
    ;
    Rademacher, Jens
    Patterns are ubiquitous in nature and can arise in reaction-diffusion systems with differential diffusions. The existence and stability of (in)homogeneous steady state are the classical topics in the dynamics of reaction-diffusion systems. In this thesis we study the influences of anomalous diffusion and advection upon the patterns. In the first part of this thesis we consider the impact of subdiffusive process on the instability of homogeneous states in three types of reaction-subdiffusion systems. The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time-fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalised spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing-type instabilities. These results are numerically illustrated. Moreover, we prove expansions that imply for one class of reaction-subdiffusion equations algebraic decay for stable spectrum, whereas for another class this is exponential. We also study the linearisation of a nonlinear reaction subdiffusion equation in a nonzero homogeneous state. Here the spectrum cannot be analysed directly by Fourier-Laplace transform, so we provide an energy estimate, existence, uniqueness and dynamics of Fourier modes of such a linearisation. It is well known that for reaction-diffusion systems with differential isotropic diffusions, a Turing instability yields striped solutions. In the second part of this thesis we study the impact of weak anisotropy by directional advection on the stability of such solutions, and the role of quadratic nonlinearities. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using Lyapunov-Schmidt reduction, Floquet-Bloch decomposition and centre manifold reduction we derive rigorous parameter expansions for existence, stability against large-wavelength and lattice modes, respectively. This provides detailed formulae for the loci of bifurcations and stability boundaries under the influences of the advection and quadratic terms. In particular, while destabilisation of the background state is through modes perpendicular to the advection (Squire-theorem), we show that stripes can bifurcate zigzag unstably. The well known destabilising effect of quadratic terms can be counterbalanced by advection, which leads to intriguing arrangements of stability boundaries. We illustrate these results numerically by an example. Finally, we show numerical computations of these stability boundaries in the extended Klausmeier model for vegetation patterns and show stripes bifurcate stably in the presence of advection.
    Dissertation
      608  386
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    Item-typ:Veröffentlichung,
    On Inhomogeneous Domain Walls in Spintronic Nanowires
    (2026-05-18)
    Siemer, Lars
    ;
    Rademacher, Jens
    ;
    Goh, Ryan
    ;
    Rademacher, Jens
    This dissertation investigates the existence, stability, and dynamical selection of magnetic domain walls in spintronic nanowires. The primary mathematical model is the Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation, which describes magnetization dynamics in an axially symmetric ferromagnetic nanowire under the influence of an applied magnetic field and a spin-polarized current. Domain walls are treated as coherent front-type structures connecting two asymptotic magnetization states. Understanding their motion and stability is relevant both to the mathematical theory of nonlinear waves and to technological applications such as racetrack memory, where information is encoded and transported through magnetic domains. A central part of the thesis concerns the existence of inhomogeneous domain walls, whose azimuthal magnetization profile is nonconstant. Using methods from dynamical systems and bifurcation theory, the dissertation proves the existence of such walls for arbitrary constant applied magnetic fields and sufficiently small nonzero spin polarization. In addition to the usual flat domain walls, the analysis identifies a distinct class of non-flat domain walls, whose axial magnetization approaches its asymptotic state in an oscillatory manner. For parameter regimes below a field threshold determined by anisotropy, damping, and spin transfer, leading-order mechanisms are derived that determine the speed and precession frequency of both flat and non-flat walls. Numerical continuation complements the analytical results and provides a broader picture of the domain-wall families in parameter space. The corresponding results also apply to the classical Landau–Lifshitz–Gilbert equation without spin-transfer torque. The dissertation then studies the stability and dynamic selection of homogeneous and inhomogeneous domain walls. Depending primarily on the applied field and spin polarization, the asymptotic states are classified into bistable and monostable regimes. In the bistable regime, existing stability results for an explicit family of precessing domain walls are extended, and a relation between the applied field and spin polarization is obtained for standing domain walls. When this family enters the monostable regime, numerical simulations show that it first becomes convectively unstable and generates pushed fronts before eventually becoming absolutely unstable. To understand the resulting dynamics, explicit formulas for the absolute spectrum of a general class of operators are derived. Translation and rotational symmetries are related to singularities of the pointwise Green’s function, providing a linear mechanism for predicting the asymptotically selected propagation speed and precession frequency. Long-time numerical simulations using freezing and phase conditions confirm these predictions and distinguish regimes of pushed and pulled front propagation. Finally, the thesis examines the broader role of spectral instability through a case study of a coupled Fisher–KPP/Swift–Hohenberg system. This model makes it possible to distinguish remnant instability, unstable absolute spectrum, and instability associated with pinched double roots. Explicit descriptions of the absolute spectra are obtained, including a formula for the Swift–Hohenberg component, and different parameter regions are classified according to their spectral and dynamical behavior. In the remnant-instability regime, the critical invasion front remains stable despite the inability to stabilize the essential spectrum by exponential weighting. In other regimes, long-time simulations reveal faster fronts that eventually overtake the expected critical front. The thesis explains this behavior through extremely weak coupling of unstable modes, such as coupling introduced by numerical round-off errors, and introduces the absolute spreading speed as a predictor of the resulting invasion velocity. Overall, the dissertation develops a unified framework connecting existence, spectral stability, and dynamical selection of coherent structures. The domain-wall analysis establishes which coherent structures exist and how they are organized in parameter space; the stability analysis explains how the LLGS dynamics select their speed and frequency; and the remnant-instability study demonstrates both the power and the limitations of spectral-selection methods in coupled systems.
    Dissertation
      17  8
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    Stability, Bifurcations and Explicit Solutions in Geophysical Fluid Models with Simplified Backscatter
    (2022-12-19)
    Prugger, Artur
    ;
    Kühn, Christian
    ;
    Rademacher, Jens
    ;
    Rademacher, Jens
    Analytical investigation methods are used in this work in order to study a simplified form of a numerical scheme, the so-called kinetic energy backscatter scheme, which models the effects of unresolved small scales on the resolved large scales in the simulations of geophysical flows in atmosphere and oceans. This thesis is divided into three main parts, an additional introduction in the beginning about the thesis and the topics it contains as well as a short outlook in the end about further ideas for research topics based on the results of this work. In the first main part certain traveling wave functions are presented, which can be used in order to explicitly solve nonlinear fluid equations. Their properties are studied, in particular possible superposition among these functions, which also solves the underlying fluid equations. The occurrence of such explicit solutions is studied in the incompressible Euler, Navier-Stokes and rotating Boussinesq equations, as well as in more general fluid equations with certain forcing terms. In the end of this part the new contributions of the presented solutions are pointed out and a comparison of them with already known ones is provided. In the second main part the rotating shallow water and Boussinesq equations with simplified kinetic energy backscatter and hyperviscosity are analyzed in a continuous setting. For these investigations the functions presented in the first part are used and numerous solutions that grow exponentially and unboundedly in time are found, which indicates the possibility of undesired energy concentration into specific modes due to the backscatter. The stability of trivial as well as certain nontrivial steady solutions are studied and the occurrence of unboundedly growing unstable perturbations shown in certain cases. In the third main part the rotating shallow water equations with backscatter, hyperdiffusion and linear as well as non-smooth quadratic bottom drag terms are studied. The provided stability analysis shows, that by decreasing the linear bottom drag the trivial flow becomes unstable after a certain threshold, which generates nonlinear flows. These nonlinear flows are investigated in more details. For isotropic backscatter and hyperdiffusion the simultaneous supercritical bifurcation of (steady) Rossby waves and (temporally oscillating) inertia-gravity waves is proved, while in the anisotropic case only Rossby waves primarily bifurcate. The bifurcation results are illustrated by numerical computations and branches are extended in parameter space beyond the analytical investigations. Furthermore, it is shown that purely smooth bottom drag cannot completely suppress the occurrence of explicit solutions as presented in the second part, so that steady and unboundedly growing explicit flows can exist in this case as well.
    Dissertation
      268  410
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    Ergodic theory of nonlinear waves in discrete and continuous excitable media
    (2021-08-25)
    Ulbrich, Dennis
    ;
    Melbourne, Ian
    ;
    Rademacher, Jens
    ;
    Rademacher, Jens
    In this thesis, we analyze discrete and continuous models of excitable media with the intention to reveal similarities between both approaches in terms of wave propagation and interaction. While the discrete perspective is represented by the one-dimensional Greenberg-Hastings cellular automata (GHCA), as a continuous model we consider the $\theta$-equations which are basic partial differential equations (PDE) for pure phase dynamics. On the one hand, qualitatively, collision and annihilation of waves can be observed in both models in striking resemblance. However, on the other hand, it turns out that a quantitative comparison of discrete and continuous wave interactions is limited due to weak wave interactions in the PDE. Specifically, complexity considerations show that a direct comparison of discrete and continuous strong wave interactions is problematic.
    Dissertation
      538  680
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    Bifurcation Analysis for Systems with Piecewise Smooth Nonlinearity and Applications
    (2021-12-17)
    Steinherr Zazo, Miriam
    ;
    Champneys, Alan
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    Rademacher, Jens
    ;
    Rademacher, Jens
    In the first part of this thesis, Andronov-Hopf bifurcations in systems with piecewise smooth nonlinearity are analyzed, which are motivated by models that arise in controlled ship maneuvering. In particular, within this nonsmooth setting, we derive explicit formulas for the generalization of the first Lyapunov coefficient, which determines the direction of branching. In addition, we show that, in general, this differs from any fixed smoothing of the vector field. Specifically, we focus on nonsmooth nonlinearities of the form the product of one variable times the absolute value of another variable. However, our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part. Furthermore, other bifurcations occur in systems with the aforementioned nonsmooth nonlinearity. In particular, we perform an analysis of normal forms with piecewise smooth nonlinear part for coefficients for Bogdanov-Takens points, and compare these results with the corresponding smooth version. After the most theoretical part of this manuscript, we consider a particular ship maneuvering model, and show that the previously developed outcomes apply in actual systems. We determine the criticality of Hopf bifurcations that arises in stabilizing the straight motion of a marine craft model. For such a 3 degree of freedom system of ship motion with yaw damping and yaw restoring control, we present a detailed study of the possibilities for stabilizing the straight motion and the resulting nonlinear effects. To facilitate the analysis, we consider a combination of rudder and propeller forces into an effective thruster force. We identify the existence, location and geometry of the stability boundary in terms of the controls, including the dependence on the propeller diameter and the thruster position. We find numerically that “safe” supercritical Hopf bifurcations are typical and, by means of numerical continuation, we provide a global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits. We illustrate the resulting stable ship motions in Earth-fixed coordinates and present some direct numerical simulations.
    Dissertation
      360  297