# Matter-wave solutions in the Bose-Einstein condensates with the harmonic and Gaussian potentials

## TL;DR

This paper derives exact matter-wave solutions for the quasi-one-dimensional Gross-Pitaevskii equation with combined harmonic and Gaussian potentials, analyzing their stability and potential experimental relevance.

## Contribution

It introduces new exact solutions for the GP equation with modulated potentials and nonlinearity, including stability analysis and parameter regimes.

## Key findings

- Several families of exact solutions are identified.
- Some solutions are found to be stable through numerical analysis.
- Parameter regimes for stability are characterized.

## Abstract

We study exact solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the (space, time)-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation, we report several families of exact solutions of the GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.