# On nonlinear profile decompositions and scattering for a NLS-ODE model

## TL;DR

This paper studies a combined NLS-ODE Hamiltonian system, demonstrating scattering for small radial solutions using Nakanishi's framework and Fermi Golden Rule estimates, advancing understanding of soliton dynamics.

## Contribution

It extends scattering results to a simplified NLS-ODE model around solitons, employing novel analytical techniques based on Nakanishi's approach.

## Key findings

- Proves scattering for small $L^2$-based $H^1$ radial solutions.
- Utilizes Fermi Golden Rule estimates within Nakanishi's framework.
- Provides a simplified model for analyzing soliton behavior in NLS systems.

## Abstract

In this paper, we consider a Hamiltonian system combining a nonlinear Schr\" odinger equation (NLS) and an ordinary differential equation (ODE). This system is a simplified model of the NLS around soliton solutions. Following Nakanishi \cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on $L^4$ in time norms.