# Self-Localized Solutions of the Nonlinear Quantum Harmonic Oscillator

## TL;DR

This paper investigates the existence, properties, and stability of self-localized soliton solutions in the nonlinear quantum harmonic oscillator using spectral renormalization and split-step Fourier methods, revealing stability conditions and pulsating behaviors.

## Contribution

It introduces a detailed analysis of single, dual, and triple soliton solutions in the NQHO, highlighting stability criteria and dynamic behaviors not previously characterized.

## Key findings

- Single and dual solitons satisfy stability conditions.
- Triple soliton solutions are unstable within studied parameters.
- Solitons exhibit pulsating dynamics during evolution.

## Abstract

We analyze the existences, properties, and stabilities of the self-localized solutions of the nonlinear quantum harmonic oscillator (NQHO) using spectral renormalization method (SRM). We show that self-localized single, dual and triple soliton solutions of the NQHO do exists, however, only single and dual soliton solutions satisfy the necessary Vakhitov and Kolokolov slope condition, therefore triple soliton solution is found to be unstable, at least for the parameter ranges considered. Additionally, we investigate the stability characteristics of the single and dual soliton solutions using a split-step Fourier scheme. We show that single and dual soliton solutions are pulsating during time stepping. We discuss our findings and comment on our results.