On the Existence of Self-Similar solutions for some Nonlinear Schr\"odinger equations

We construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are stable. As such they are not of the standard asymptotic decomposition of linear waves and localized waves. Such weakly localized solutions were expected in view of previous works \cite{Liu-Sof1,Liu-Sof2} 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''.