  We construct solutions of Schr\"odinger equations which have asymptotic self
similar solutions as time goes to infinity. Also included are situations with
two-bubbles. These solutions are global, with constant non-zero $L^2$ norm, and
are stable. As such they are not of the standard asymptotic decomposition of
linear wave and localized waves. Such weakly localized waves were expected in
view of previous works on the large time behavior of general dispersive
equations. It is shown that one can associate a \emph{scattering channel} to
such solutions, with the Dilation operator as the asymptotic "hamiltonian".
