On the asymptotic stability on the line of ground states of the pure power NLS with $ 0\le 2-p \ll 1 $

TL;DR

This paper investigates the asymptotic stability of ground states for the pure power nonlinear Schrödinger equation on the line, focusing on exponents close to 2 and addressing challenges due to nonlinearity regularity loss.

Contribution

It extends previous work by analyzing stability for exponents near 2, incorporating spectral results from computational studies, and developing new methods for handling nonlinearity regularity issues.

Findings

Confirmed asymptotic stability for exponents close to 2

Identified spectral properties influencing stability

Developed new analytical techniques for low-regularity nonlinearities

Abstract

We continue our series devoted, after references \cite{CM24D1} and \cite{CM243}, at proving the asymptotic stability of ground states of the pure power Nonlinear Schr\"odinger equation on the line. Here we assume some results on the spectrum of the linearization obtained computationally by Chang et al. \cite{Chang} and then we explore the equation for exponents $p\le 2$ sufficiently close to 2. The ensuing loss of regularity of the nonlinearity requires new arguments.