On small energy solutions of the Nonlinear Schr\"odinger Equation in 1D with a generic trapping potential with a single eigenvalue

TL;DR

This paper establishes results on small energy solutions of the 1D nonlinear Schrödinger equation with a trapping potential, extending previous work to a broader range of nonlinear exponents using virial inequalities and smoothing estimates.

Contribution

It generalizes existing results to cover all power nonlinearities with p>1 in one dimension, employing new analytical techniques.

Findings

Proves stability and decay properties for small solutions

Extends the range of nonlinear exponents analyzed

Utilizes virial inequalities and smoothing estimates effectively

Abstract

We prove in dimension $d=1$ a result similar to Soffer and Weinstein Jour. Diff. Eq. 98 (1992) capturing for pure power nonlinearities the whole range of exponents $p>1$. The proof is based on the virial inequality of Kowalczyk \textit{{et al.}} J. Eur. Math. Soc. (JEMS) 24 (2022) with smoothing estimates like in Mizumachi J. Math. Kyoto Univ. 48 (2008).