  We discuss stochastic phase-space methods within the truncated Wigner
approximation and show explicitly that they can be used to solve
non-equilibrium dynamics of bosonic atoms in one-dimensional traps. We consider
systems both with and without an optical lattice, and address different
approximations in the stochastic synthesization of quantum statistical
correlations of the initial atomic field. We also present a numerically
efficient projection method for analyzing correlation functions of the
simulation results, and demonstrate physical examples of non-equilibrium
quantum dynamics of solitons and atom number squeezing in optical lattices.
