Magnetic wave dynamics in ferromagnetic thin films: Interactions of solitons and positons in Landau–Lifshitz–Gilbert equation

Abstract

This study explores the higher-order soliton and smooth positon solutions derived from the Landau–Lifshitz–Gilbert (LLG) equation which describes the wave propagation in the physical setting of ferromagnetic thin films by adopting the Darboux transformation (DT) technique. The LLG equation is reformulated into an integrable nonlinear Schrödinger-like equation under the long-wave approximation. This approach facilitates the analysis of the propagation and interaction of magnetic solitons and positons within a ferromagnetic thin films, by taking into account the effects of interfacial Dzyaloshinskii–Moriya (DM) interaction. The DM interaction notably influences the soliton and positon velocities, alters their collision positions, and impacts their propagation trajectories. The magnetization dynamics, described by $\overrightarrow{m}=m_x{\overrightarrow{e}}_x+m_y{\overrightarrow{e}}_y+m_z{\overrightarrow{e}}_z$, exhibits distinct characteristics where $m_x$ and $m_y$ represent oscillations in the plane perpendicular to the external field with breather-like properties, while $m_z$ forms localized magnetized states. The magnetization dynamics are characterized through Bloch sphere trajectories, revealing progressively localized spiral patterns for first- and second-order soliton and positon solutions, indicating enhanced spatial confinement of magnetic moments. Moreover, the dynamic behavior of smooth positons within the NLS-type equation is studied using the decomposition method of the modulus square. This methodology offers an approximate characterization of positon trajectories and the time-dependent “phase shift” occurring after collisions.