  We study semiclassical dynamics of anisotropic Heisenberg models in two and
three dimensions. Such models describe lattice spin systems and hard core
bosons in optical lattices. We solve numerically Landau-Lifshitz type equations
on a lattice and show that in the phase diagram of magnetization and
interaction anisotropy, one can identify several distinct regimes of dynamics.
These regions can be distinguished based on the character of one dimensional
solitonic excitations, and stability of such solitons to transverse modulation.
Small amplitude and long wavelength perturbations can be analyzed analytically
using mapping of non-linear hydrodynamic equations to KdV type equations.
Numerically we find that properties of solitons and dynamics in general remain
similar to our analytical results even for large amplitude and short distance
inhomogeneities, which allows us to obtain a universal dynamical phase diagram.
As a concrete example we study dynamical evolution of the system starting from
a state with magnetization step and show that formation of oscillatory regions
and their stability to transverse modulation can be understood from the
properties of solitons. In regimes unstable to transverse modulation we observe
formation of lump type solutions with modulation in all directions. We discuss
implications of our results for experiments with ultracold atoms.

Quantum Gases;Strongly Correlated Electrons;Quantum Physics