On scattering of small energy solutions of non autonomous hamiltonian nonlinear Schr\"odinger equations

TL;DR

This paper proves that small energy solutions to a non-autonomous cubic nonlinear Schrödinger equation with a localized potential tend to free solutions over time, extending previous results using Hamiltonian structure techniques.

Contribution

It demonstrates asymptotic freedom for small solutions of a non-autonomous NLS with potential, under generic linearization conditions, advancing the Hamiltonian approach in this context.

Findings

Small energy solutions are asymptotically free.

Extension of Hamiltonian methods to non-autonomous NLS.

General conditions ensuring scattering behavior.

Abstract

We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schr\" odinger equation (NLS) with an attractive localized potential and a time-dependent factor in the nonlinearity. We show that, under generic hypotheses on the linearization at 0 of the equation, small energy solutions are asymptotically free. This is yet a new application of the hamiltonian structure, continuing a program initiated in a paper by Bambusi and Cuccagna.