Mathematical theory and numerical methods for Bose-Einstein condensation

TLDR

The paper reviews recent mathematical theory and numerical methods for Bose‑Einstein condensation based on the Gross‑Pitaevskii equation, compares physical and semiclassical scalings in the Thomas‑Fermi regime, discusses semiclassical limits, and outlines future models for spin‑1, Bogoliubov, and finite‑temperature BEC. It analyzes GPE reductions, ground‑state existence and dynamics, and reviews numerical schemes—including gradient‑flow with discrete normalization, finite‑difference, and time‑splitting spectral methods—providing error estimates and extensions to rotating, dipolar, and multi‑component condensates. The authors show that numerical methods simplify for symmetric GPEs and provide insights into semiclassical limits under the two scalings.

Abstract

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.