Long time oscillation of solutions of nonlinear Schr\"odinger equations near minimal mass ground state

TL;DR

This paper investigates the long-term oscillatory behavior of radially symmetric solutions to nonlinear Schrödinger equations near the minimal mass ground state, revealing a finite-dimensional approximation and separation from the infinite-dimensional dynamics.

Contribution

It introduces a new coordinate system near the minimal mass ground state and demonstrates long time oscillations through finite and infinite dimensional analysis.

Findings

Existence of solutions oscillating for long times near the minimal mass ground state.

Finite dimensional dynamics approximate Newton's equations in an anharmonic potential.

Infinite dimensional part remains well separated, enabling long-term oscillations.

Abstract

In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schr\"odinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time. More precisely, we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator. Then, we show that the finite dimensional part, two dimensional, approximately obeys Newton's equation of motion for a particle in an anharmonic potential well. Showing that the infinite dimensional part is well separated from the finite dimensional part, we will have long time oscillation.