  We study the asymptotic behaviour in time of solutions and the theory of
scattering for the modified Schr"odinger map in two space dimensions. We solve
the Cauchy problem with large finite initial time, up to infinity in time, and
we determine the asymptotic behaviour in time of the solutions thereby
obtained. As a byproduct, we obtain global existence for small data in H^k
inter FH^k with k > 1. We also solve the Cauchy problem with infinite initial
time, namely we construct solutions defined in a neighborhood of infinity in
time, with prescribed asymptotic behaviour of the previous type.
