Global dynamics above the first excited energy for the nonlinear Schr\"odinger equation with a potential

TL;DR

This paper classifies the long-term behavior of solutions to a focusing nonlinear Schrödinger equation with a potential, especially those near the first excited energy level, revealing stable, unstable, and blow-up dynamics.

Contribution

It provides a detailed classification of global dynamics above the first excited energy for the nonlinear Schrödinger equation with potential, including stability and blow-up scenarios.

Findings

Classified solutions into 9 distinct sets based on energy and symmetry.

Identified a stable set approaching ground states with large radiation.

Described blow-up behavior for solutions near the first excited solitons.

Abstract

Consider the focusing nonlinear Schr\"odinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.