  We study the Gross-Pitaevskii equation in dimension two with periodic
conditions in one direction, or equivalently on the product space $\mathbb{R}
\times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L
\mathbb{Z}.$ We focus on the variational problem consisting in minimizing the
Ginzburg-Landau energy under a fixed momentum constraint. We prove that there
exists a threshold value for $L$ below which minimizers are the one-dimensional
dark solitons, and above which no minimizer can be one-dimensional.
