Global dynamics below the ground states for NLS under partial harmonic confinement

TL;DR

This paper investigates the long-term behavior of solutions to a focusing nonlinear Schrödinger equation with partial harmonic confinement, establishing precise initial data conditions for blow-up or scattering outcomes.

Contribution

It provides a necessary and sufficient criterion for global behavior below ground states, using variational methods and profile decompositions.

Findings

Characterization of initial data for global existence or blow-up

Conditions for scattering established below ground states

Application of virial estimates and profile decompositions

Abstract

We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{\"o}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.