  In this letter we study the non-equilibrium spin dynamics in the
non-integrable spin-1 XXZ chain, emerging after joining two macroscopic pure
states with different magnetizations. Employing the so-called time-dependent
variational principle (TDVP) we simulate the dynamics of the total
magnetization in the right (left) half of the system up to times that are
normally inaccessible by standard tDMRG methods. We identify three distinct
phases depending on the anisotropy of the chain, corresponding to diffusive,
marginally super-diffusive and insulating. We observe a transient ballistic
behaviour with a crossover time that diverges as the isotropic point is
approached. We conclude that on intermediate-large time scales the dynamics is
well described by the integrable Landau-Lifschitz classical differential
equation.
