A normalized gradient flow method for computing ground states of spin-2 Bose-Einstein condensates

TL;DR

This paper introduces a novel normalized gradient flow method tailored for accurately computing the ground states of spin-2 Bose-Einstein condensates, addressing the complexities of multiple components and physical constraints.

Contribution

The paper extends the normalized gradient flow technique to spin-2 BECs by designing a new projection method that handles multiple constraints and proves solution uniqueness.

Findings

The method efficiently computes ground states with high accuracy.

Numerical results reveal diverse physical phenomena in spin-2 BECs.

The approach is robust across different phases and potentials.

Abstract

We propose and analyze an efficient and accurate numerical method for computing ground states of spin-2 Bose-Einstein condensates (BECs) by using the normalized gradient flow (NGF). In order to successfully extend the NGF to spin-2 BECs which has five components in the vector wave function but with only two physical constraints on total mass conservation and magnetization conservation, two important techniques are introduced for designing the proposed numerical method. The first one is to systematically investigate the ground state structure and property of spin-2 BECs within a spatially uniform system, which can be used on how to properly choose initial data in the NGF for computing ground states of spin-2 BECs. The second one is to introduce three additional projection conditions based on the relations between the chemical potentials, together with the two existing physical constraints, such that the five projection parameters used in the projection step of the NGF can be uniquely determined. Then a backward-forward Euler finite difference method is adapted to discretize the NGF. We prove rigorously that there exists a unique solution of the nonlinear system for determining the five projection parameters in the full discretization of the NGF under a mild condition on the time step size. Extensive numerical results on ground states of spin-2 BECs with different types of phases and under different potentials are reported to show the efficiency and accuracy of the proposed numerical method and to demonstrate several interesting physical phenomena on ground states of spin-2 BECs.