Global dynamics below excited solitons for the nonlinear Schr\"odinger equation with a potential

TL;DR

This paper classifies the long-term behavior of solutions to a nonlinear Schrödinger equation with potential, focusing on the dynamics below the second lowest energy solitons under small mass and radial symmetry.

Contribution

It provides a classification of global dynamics for NLS with potential below the second lowest energy solitons, considering stability and instability regimes.

Findings

Stable solitons with negative small energy are asymptotically stable.

Unstable solitons with positive large energy exist in the focusing case.

The dynamics below the second lowest energy are fully characterized under certain constraints.

Abstract

Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.