  We investigate the Dirac time-dependent variational method using a Gaussian
trial functional for an infinite one dimensional system of Bosons interacting
through a repulsive contact interaction. The method produces a set of
non-linear time dependent equations for the variational parameters. By solving
the static equations we have calculated the ground state energy per particle.
We have also considered small oscillations about equilibrium and obtain mode
equations which lead us to a gapless dispersion relation. The existence of an
exact numerical solution for the ground state energy and excitations obtained
by Lieb allow us to compare with the Gaussian results. We can also, as the
system becomes less dilute, see the improvement of the results as compared with
the Bogoliubov scheme.
