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.