# Ground State Solutions of the Complex Gross Pitaevskii Equation Associated to Exciton-Polariton Bose-Einstein Condensates

## TL;DR

This paper proves the existence of ground state solutions for a Gross-Pitaevskii equation modeling exciton-polariton Bose-Einstein condensates, highlighting their macroscopic quantum nature and addressing mathematical challenges from pumping and damping terms.

## Contribution

It introduces a novel mathematical analysis demonstrating ground state existence for a complex Gross-Pitaevskii equation with pumping and damping effects.

## Key findings

- Existence of ground state solutions established
- Condensation occurs under certain conditions
- Solved the associated Cauchy problem and derived laws

## Abstract

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schroedinger equation and prove some corresponding laws.