Compact Darboux transformation and multi-pole magnetic waves within a higher-order Heisenberg ferromagnetic model

Spin waves, as charge-free collective excitations of electron spins, are fundamental to the development of energy-efficient spintronic devices and wave-based computing. This Letter investigates a higher-order Heisenberg ferromagnetic model characterizing the dynamics of the magnetic vector in isotropic ferromagnetism. We construct a direct $N$-fold Darboux transformation (DT) and a generalized $N$-fold DT in the compact determinant forms, which enhance the efficiency of obtaining rogue wave, multiple, multi-pole and mixed solutions for the nonlinear systems within the Heisenberg ferromagnetic hierarchy. Those results also help provide a mathematical framework for exploring complex spin wave interactions in ferromagnetic materials.