# Soliton-potential interactions for nonlinear Schr\"odinger equation in $\mathbb{R}^3$

## TL;DR

This paper investigates the complex dynamics and scattering behavior of narrow solitons in the nonlinear Schrödinger equation in three-dimensional space, especially how interactions with potentials can significantly alter their velocities over time.

## Contribution

It provides new insights into soliton-potential interactions, demonstrating that the outgoing soliton velocity can differ greatly from the initial velocity, extending previous stability results.

## Key findings

- Soliton velocity can change significantly after interaction with a potential.
- The asymptotic state of the system can be far from the initial state.
- Previous stability results assume small velocity changes, which this work challenges.

## Abstract

In this work we mainly consider the dynamics and scattering of a narrow soliton of NLS equation with a potential in $\mathbb{R}^3$, where the asymptotic state of the system can be far from the initial state in parameter space. Specifically, if we let a narrow soliton state with initial velocity $\upsilon_{0}$ to interact with an extra potential $V(x)$, then the velocity $\upsilon_{+}$ of outgoing solitary wave in infinite time will in general be very different from $\upsilon_{0}$. In contrast to our present work, previous works proved that the soliton is asymptotically stable under the assumption that $\upsilon_{+}$ stays close to $\upsilon_{0}$ in a certain manner.