Global dynamics below excited solitons for the non-radial NLS with potential

TL;DR

This paper studies the long-term behavior of solutions to the 3D cubic nonlinear Schrödinger equation with an external potential, showing that small solutions either scatter or grow, extending previous radial results to non-radial cases.

Contribution

It extends the analysis of global dynamics for the NLS with potential from radial to non-radial solutions at small mass.

Findings

Solutions with energy below excited solitons either scatter or grow in $H^1$ norm.

The result generalizes Nakanishi's radial case to non-radial solutions.

Provides a comprehensive understanding of solution behavior in the presence of external potential.

Abstract

We consider the global dynamics of solutions to the $3d$ cubic nonlinear Schr\"odinger equation in the presence of an external potential, in the setting in which the equation admits both ground state solitons and excited solitons at small mass. We prove that small mass solutions with energy below that of the excited solitons either scatter to the ground states or grow their $H^1$-norm in time. In particular, we give an extension of the result of Nakanishi [19] from the radial to the non-radial setting.