Few-magnon excitations in Heisenberg-like models play an important role in understanding magnetism and have long been studied by various approaches. However, the quantum dynamics of magnon excitations in a finite-size spin-$S$ $XXZ$ chain with single-ion anisotropy remains unsolved. Here, we exactly solve the two-magnon (three-magnon) problem in the spin-$S$ $XXZ$ chain by reducing the few-magnons to a fictitious single particle on a one-dimensional (two-dimensional) effective lattice. Such a mapping allows us to obtain both the static and dynamical properties of the model explicitly. The zero-energy-excitation states and various types of multimagnon bound states are manifested, with the latter being interpreted as edge states on the effective lattices. Moreover, we study the real-time multimagnon dynamics by simulating single-particle quantum walks on the effective lattices.
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