  We consider the non-equilibrium dynamics of a weakly interacting Bose gas
tightly confined to a highly elongated double well potential. We use a
self-consistent time-dependent Hartree--Fock approximation in combination with
a projection of the full three-dimensional theory to several coupled
one-dimensional channels. This allows us to model the time-dependent splitting
and phase imprinting of a gas initially confined to a single quasi
one-dimensional potential well and obtain a microscopic description of the
ensuing damped Josephson oscillations.
