In the field-driven DW case, it is rigorously proven that a static DW solution is not possible to exist in a homogeneous ferrimagnetic nanowire under a uniform external magnetic field that makes the energy densities in two domains different. The spins inside the DW texture must vary over time under the external field, leading to energy dissipation because of the Gilbert damping mechanism. This energy dissipation is compensated by the release of Zeeman energy from shifting the DW toward the higher energy density domain. The origin of fast DW motion and the divergence of Walker breakdown field at the AMCP are then explained. An analytical expression of DW velocity under fields higher than the Walker breakdown field is derived, consistent with existing experimental and simulation results.

In the case of a current-driven DW, our study investigates the effects of a spin-polarized electric current passing through a DW in a ferrimagnetic nanowire. This current generates two types of spin transfer torque (STT) components: a larger component called adiabatic spin transfer torque (a-STT) and a smaller non-adiabatic spin transfer torque (na-STT). Our research demonstrates that a-STT can be included into the Lagrangian as an energy functional, and tends to induce a twisting of the DW planes. Notably, DWs in homogeneous ferrimagnetic nanowires can resist a-STT until it reaches a threshold strength proportional to the maximal transverse field arising from transverse anisotropy. In contrast, na-STT cannot be included in the Lagrangian and is introduced through the Rayleigh dissipation functional. We rigorously prove that a static DW solution is impossible to exist under the influence of even an infinitesimally small na-STT. The motion of DWs is governed by the energy-work principle, with STTs performing positive work to compensate for dissipated energy. When the applied current density is lower than the Walker breakdown current density, only na-STT performs positive work (a-STT does no work). However, when the applied current density is higher than the Walker breakdown current density, DWs begin to exhibit a precessional motion around the FiM nanowire axis. The direction of this precession determines whether a-STT performs positive or negative work, while na-STT continues to perform positive work. A formula for DW velocity derived from the energy-work principle aligns with simulations conducted for all range of cuurent densities. Notably, near the AMCP, the DW precession frequency reaches a peak due to the influence of a-STT. This significantly distorts the DW structure and modifies its motion, leading to deviations from the linear current density dependence predicted by theories without considering DW plane twisting. Our theoretical framework effectively explains the observed DW mobility near the AMCP in ferrimagnetic nanowires and resolves the long-standing issue of the unphysical negative na-STT problem.

Recently, magnetic skyrmions attracts more attention due to their promising applications as information carries in magnetic nano-devices. Our research on skyrmions focuses on inhomogeneous thin film chiral magnets, which are significant for both practical applications and fundamental interests, as inhomogeneity can induce exotic physics absent in homogeneous samples.

The first part of the skyrmion study explores ferromagnetic skyrmion pinning by different kinds of disk-shaped defects. Precise skyrmion pinning, achieved through intentionally designed magnetic nanostructures, is essential for an effective control of skyrmion. This study differs large-η skyrmion (with a large skyrmion core and a relatively thin skyrmion wall) and small-η skyrmions (with a negligible skyrmion core and a relatively thick skyrmion wall) based on their pinning positions and potential landscapes. It reveals that skyrmion walls favor regions with lower exchange stiffness, higher DMI constant, or lower magnetic anisotropy, while skyrmion cores are solely influenced by the perpendicular magnetic anisotropy. Depending on the type of defect, type of skyrmion, and relative defect size, skyrmions can exhibit symmetrical or asymmetrical pinning. Thermal agitation can dislodge skyrmions pinned by local energy minima, with the pinning lifetime adhering to the Néel-Arrhenius law. Notably, when the disk size is close to the skyrmion size, skyrmions undergo great deformation (shrinkage or expansion) to fill the entire disk.

The second part of the skyrmion study explores current-driven antiferromagnetic (AFM) skyrmion dynamics in disordered thin films. Investigating skyrmion motion in inhomogeneous systems is an important subject since the material disorders are impossible to avoid in real material systems. Micromagnetic simulations reveal that AFM skyrmions deviate from their intended trajectories in homogeneous thin films due to the presence of disorder. Specifically, the simulations demonstrate a Brownian motion in the transverse direction, while longitudinal motion is slowed by the disorder. An effective theory using the stochastic Thiele equation explains these observations. We also find that at strong disorders above a critical value dependent on current density, AFM skyrmions become pinned.

Overall, this thesis deepens the understanding of the statics and dynamics of DWs and skyrmions in ferromagnetic, ferrimagnetic, and AFM materials, providing valuable insights for designing future spintronic devices based on topological magnetic solitons.