  Motivated by transverse stability issues, we address the time evolution under
the KP-II flow of perturbations of a solution which does not decay in all
directions, for instance the KdV-line soliton. We study two different types of
perturbations : perturbations that are square integrable in $ \R\times \T $ and
perturbations that are square integrable in $ \R^2 $. In both cases we prove
the global well-posedness of the Cauchy problem associated with such initial
data.
