  Motivated by recent experimental progress in the quasi-one-dimensional
quantum magnet NiNb$_2$O$_6$, we study the spin dynamics of an S=1
ferromagnetic Heisenberg chain with single-ion anisotropy by using a
semiclassical molecular dynamics approach. This system undergoes a quantum
phase transition from a ferromagnetic to a paramagnetic state under a
transverse magnetic field, and the magnetic responses reflecting this
transition is well described by our semiclassical method. We show that at
low-temperature the transverse component of the dynamical structure factor
depicts clearly the magnon dispersion, and the longitudinal component exhibits
two continua associated with single- and two-magnon excitations, respectively.
These spin excitation spectra show interesting temperature dependence as
effects of magnon interactions.Our findings shed light on experimental
detection of spin excitations in a large class of quasi-one-dimensional
magnets.
