# Decay of small odd solutions for long range Schr\"odinger and Hartree equations in one dimension

## TL;DR

This paper studies the long-term decay of odd solutions to one-dimensional nonlinear Schrödinger and Hartree equations, demonstrating decay in various nonlinear regimes using virial identities without spectral assumptions.

## Contribution

It provides new decay results for odd solutions of NLS and Hartree equations in one dimension, covering sub, critical, and supercritical nonlinearities with a unified virial identity approach.

## Key findings

- Decay to zero in compact regions over time for various nonlinearities
- Applicable to NLS with potential and Hartree equations without spectral assumptions
- Results cover a wide range of nonlinear regimes, including long-range cases

## Abstract

We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schr\"odinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space $H^1(\mathbb{R})$ only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models as in Kowalczyk-Martel-Mu\~noz, and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.