  We consider the Schr{\"o}dinger equation with a logarithmic nonlinearty and
non-trivial boundary conditions at infinity. We prove that the Cauchy problem
is globally well posed in the energy space, which turns out to correspond to
the energy space for the standard Gross-Pitaevskii equation with a cubic
nonlinearity, in small dimensions. We then characterize the solitary and
travelling waves in the one dimensional case.
