  We consider the nonlinear Schr\"odinger equation with nonzero conditions at
infinity in $\R^2$. We investigate the existence of traveling waves that are
periodic in the direction transverse to the direction of propagation and
minimize the energy when the momentum is kept fixed. We show that for any given
value of the momentum, there is a critical value of the period such that
traveling waves with period smaller than the critical value are
one-dimensional, and those with larger periods depend on two variables.
