  Results are presented for the nonequilibrium dynamics of a quantum $XXZ$-spin
chain whose spins are initially arranged in a domain wall profile via the
application of a magnetic field in the $z$-direction which is spatially varying
along the chain. The system is driven out of equilibrium in two ways: a). by
rapidly turning off the magnetic field, b). by rapidly quenching the
interactions at the same time as the magnetic field is turned off. The
time-evolution of the domain wall profile as well as various two-point spin
correlation functions is studied by the exact solution of the fermionic problem
for the $XX$ chain and via a bosonization approach and a mean-field approach
for the $XXZ$ chain. At long times the magnetization is found to equilibrate
(reach the ground state value), while the two-point correlation functions in
general do not. In particular, for quenches within the gapless $XX$ phase, the
transverse spin correlation functions acquire a spatially inhomogeneous
structure at long times whose details depend on the initial domain wall
profile. The spatial inhomogeneity is also recovered for the case of classical
spins initially arranged in a domain wall profile and shows that the
inhomogeneities arise due to the dephasing of transverse spin components as the
domain wall broadens. A generalized Gibbs ensemble approach is found to be
inadequate in capturing this spatially inhomogeneous state.
