Global Self-Similar solutions, Breakdown of Asymptotic Completeness-New Physics?

For Schrödinger-type equations with time-dependent, spatially local interactions — including nonlinear cases — there exist solutions that are neither localized nor scattering. These solutions exhibit sub-ballistic spreading, yet are global, asymptotically stable, and transport mass (the L^2-norm) to infinity. I will describe the construction of such self-similar solutions and discuss some of their physical implications.