The asymptotic stability on the line of ground states of the pure power NLS with $0<|p-3|\ll 1$

TL;DR

This paper proves the asymptotic stability of ground states for the one-dimensional pure power nonlinear Schrödinger equation when the exponent is close to 3, focusing on spatially even solutions and adapting existing methods.

Contribution

It extends the stability analysis to exponents near 3, modifying previous arguments by replacing virial inequalities with smoothing estimates for the initial data.

Findings

Ground states are asymptotically stable for exponents close to 3.

The proof adapts virial inequality techniques with smoothing estimates.

Results are specific to spatially even solutions on the line.

Abstract

For exponents $p$ satisfying $0<|p-3|\ll 1$ and only in the context of spatially even solutions we prove that the ground states of the nonlinear Schr\"odinger equation (NLS) with pure power nonlinearity of exponent $p$ in the line are asymptotically stable. The proof is similar to a related result of Martel, preprint arXiv:2312.11016, for a cubic quintic NLS. Here we modify the second part of Martel's argument, replacing the second virial inequality for a transformed problem with a smoothing estimate on the initial problem, appropriately tamed by multiplying the initial variables and equations by a cutoff.