On Inhomogeneous Domain Walls in Spintronic Nanowires

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.