Ground states and droplet regimes of the extended Gross-Pitaevskii equation with Lee-Huang-Yang correction

TL;DR

This paper investigates the ground states of an extended Gross-Pitaevskii equation with Lee-Huang-Yang correction, deriving reduced models, establishing existence results, and exploring different regimes through numerical simulations.

Contribution

It provides a comprehensive analysis combining theoretical derivations and numerical methods to understand ground states and regimes in the extended Gross-Pitaevskii equation.

Findings

Existence and nonexistence results for ground states in various dimensions.

Identification of different regimes including no-ground-state, soliton-like, and droplet-like.

Introduction of a flat-top approximation for droplet regimes.

Abstract

We study the ground states of the extended Gross--Pitaevskii equation with the Lee--Huang--Yang correction from both theoretical and numerical perspectives. Starting from the three-dimensional model, we derive reduced one- and two-dimensional equations through nondimensionalization and dimensional reduction. We establish existence and nonexistence results for ground states in different spatial dimensions, both in free space and under confining external potentials. For the numerical computation of ground states, we propose a normalized gradient flow method with a Lagrange multiplier. The numerical results show how the model parameters affect the ground-state profiles, and reveal different regimes in the free-space parameter plane, including no-ground-state, soliton-like, and droplet-like regions. We also introduce a simple flat-top approximation for the droplet regime and present two- and three-dimensional computations to illustrate more general localized structures.