  We review some recent results on the theory of scattering and more precisely
on the local Cauchy problem at infinity in time for some long range nonlinear
systems including some form of the Schr"odinger equation. We consider in
particular the Wave-Schr"odinger system in space dimension 3, the
Maxwell-Schr"odinger system in space dimension 3, the Klein-Gordon-Schr"odinger
system in space dimension 2 and the Zakharov system in space dimensions 2 and
3. By the use of a direct method which is intrinsically restricted to the case
of small Schr"odinger data and to the borderline long range case, one can prove
the existence of solutions defined for large times and with prescribed
asymptotic behaviour in time, without any size restriction on the Wave, Maxwell
or Klein-Gordon data. Furthermore one obtains convergence rates of the
solutions to their asymptotic forms as negative powers of t in suitable norms.
