  We investigate the properties of the one-dimensional Bose gas at zero
temperature, for which exact results exist for some model systems. We treat the
interactions between particles in the gas with an approximate form of the
many-body T-matrix, and find a form of Gross-Pitaevskii equation valid in 1D.
The results presented agree with the exact models in both the weakly and
strongly interacting limits, and interpolate smoothly between them. We also
investigate the use of mean-field treatments of trapped BECs to describe the 1D
system in the strongly interacting limit. We find that the use of the many-body
T-matrix to describe interactions leads to qualitative agreement with the exact
models for some physical quantities. We indicate how the standard mean-field
treatments need to be modified to extend and improve the agreement.
