# Nonlinear Schr\"odinger Equations for Bose-Einstein Condensates

## TL;DR

This paper analyzes the nonlinear Schrödinger equation, specifically the Gross-Pitaevskii equation, to understand Bose-Einstein condensates' behavior under various conditions through analytical and numerical methods.

## Contribution

It provides an analytical study of the NLS with $L^2$ initial data and includes numerical simulations for 2D GPE with anisotropic potentials, advancing understanding of BEC dynamics.

## Key findings

- Analytical results on wave propagation in BECs.
- Numerical simulations demonstrating wave behavior under anisotropic potentials.
- Insights into the effects of electromagnetic fields on BECs.

## Abstract

The Gross-Pitaevskii equation, or more generally the nonlinear Schr\"odinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with $L^2$ initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.