  In this note we study the initial value problem in a critical space for the
one dimensional Schr\"odinger equation with a cubic non-linearity and under
some smallness conditions. In particular the initial data is given by a
sequence of Dirac deltas with different amplitudes but equispaced. This choice
is motivated by a related geometrical problem; the one describing the flow of
curves in three dimensions moving in the direction of the binormal with a
velocity that is given by the curvature.
