The large time asymptotics of nonlinear multichannel Schroedinger equations

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear multichannel Schrödinger equations with localized, possibly time-dependent and nonlinear interactions, proving asymptotic convergence to free and localized states under certain conditions.

Contribution

It introduces a phase-space analysis and new propagation estimates to unify the treatment of linear potentials and nonlinear interactions in Schrödinger equations.

Findings

Solutions asymptotically approach free waves and localized states.

Established properties of localized solutions.

Developed a new analytical framework for nonlinear dispersive dynamics.

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.