The large time asymptotics of nonlinear multichannel Schroedinger equations

Abstract

We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev embedding. Under the assumption of radial symmetry and boundedness in H1(R3) of the solution, uniformly in time. we prove it is asymptotic in L2 (and H1) in the strong sense, to a free wave and a weakly localized solution. The general properties of the localized solutions are derived. The proof is based on the introduction of phase-space analysis of the nonlinear dispersive dynamics and relies on a new class of (exterior) a priory propagation estimates. This approach allows a unified analysis of general linear time-dependent potentials and non-linear interactions.