  Using the framework of nonlinear fluctuating hydrodynamics (NFH), we examine
equilibrium spatio-temporal correlations in classical ferromagnetic spin chains
with nearest neighbor interactions. In particular, we consider the classical
XXZ-Heisenberg spin chain (also known as Lattice Landau Lifshitz or LLL model)
evolving deterministically and chaotically via Hamiltonian dynamics, for which
energy and $z$-magnetization are the only locally conserved fields. For the
easy-plane case, this system has a low-temperature regime in which the
difference between neighboring spin's angular orientations in the XY plane is
an \textit{almost conserved} field. According to the predictions of NFH, the
dynamic correlations in this regime exhibit a heat peak and propagating sound
peaks, all with anomalous broadening. We present a detailed molecular dynamics
test of these predictions and find a reasonably accurate verification. We find
that, in a suitable intermediate temperature regime, the system shows two sound
peaks with Kardar-Parisi-Zhang (KPZ) scaling and a heat peak where the expected
anomalous broadening is less clear. In high temperature regimes of both easy
plane and easy axis case of LLL, our numerics show clear diffusive spin and
energy peaks and absence of any sound modes, as one would expect. We also
simulate an integrable version of the XXZ-model, for which the ballistic
component instead moves with a broad range of speeds rather than being
concentrated in narrower peaks around the sound speed.
