A unified framework to explore soliton boundary interaction using topological magnetic soliton spring oscillators.

Soliton-boundary interactions significantly influence the dynamics, stability, and functionality of topological magnetic solitons in spintronic devices, yet quantifying these interactions remains challenging. In this work, we introduce a unified framework termed the "topological magnetic soliton spring oscillator", designed to systematically explore and quantify soliton-boundary interactions across different soliton types, including hopfions, skyrmions, and domain walls. Within this framework, soliton motion is governed by the competition between two effective forces: the spin-transfer torque-induced driving force​, and the boundary-induced repulsive force​, which mainly arises from excess exchange energy near boundaries. Through comprehensive micromagnetic simulations, we reveal that the interaction behavior transitions between linear and nonlinear regimes, depending on the degree of soliton deformation and the value of the damping factor. Specifically, for zero damping (α = 0), when the soliton deformation is minimal, the interaction energy exhibits a linear dependence on coordinates, while significant deformation induces nonlinearity with a slope increasing toward boundaries. For small damping factors, solitons exhibit damped oscillations with velocity-dependent, multivalued interaction energy. At larger damping, overdamped dynamics dominate, characterized by nonlinear interactions with a decreasing slope as the boundary is approached. This framework not only provides a physical basis for interpreting soliton-boundary behavior in topological magnetic systems but also identifies key dynamical features relevant to the design of soliton-based spintronic devices.