Convergence analysis of Sobolev Gradient flows for the rotating Gross-Pitaevskii energy functional

TL;DR

This work addresses computing the ground state of rotating Bose–Einstein condensates by minimizing the Gross–Pitaevskii energy $E$ under the $L^2$ mass constraint on $\mathcal{M}$. It analyzes three Sobolev gradient-flow schemes, $H_0^1$, $a_0$, and $a_u$, and establishes global convergence for the $H_0^1$ and $a_0$ schemes in the rotating setting, along with local linear convergence for all three schemes in a quotient metric that accounts for phase invariance. The analysis combines energy-dissipation estimates, second-variation properties at a locally quasi-unique ground state $u^*$, and quotient-space geometry to overcome non-uniqueness induced by rotation. Numerical experiments corroborate the theory, showing that the $a_u$ scheme often yields the fastest convergence and that all schemes exhibit linear convergence with appropriate step sizes, with mesh refinement having little effect on rates. Together, the results provide constructive, scheme-specific convergence guarantees for rotating GPE gradient flows and complement broader Riemannian frameworks.

Abstract

This paper studies the numerical approximation of the ground state of rotating Bose--Einstein condensates, formulated as the minimization of the Gross--Pitaevskii energy functional under a mass conservation constraint. To solve this problem, we consider three Sobolev gradient flow schemes: the $H_0^1$ scheme, the $a_0$ scheme, and the $a_u$ scheme. Convergence of these schemes in the non-rotating case was established by Chen et al., and the rotating $a_u$ scheme was analyzed in Henning et al. In this work, we prove the global convergence of the $H_0^1$ and $a_0$ schemes in the rotating case, and establish local linear convergence for all three schemes near the ground state. Numerical experiments confirm our theoretical findings.

Figures (4)

• Figure 1: Density and phase of the ground state $u^*$.
• Figure 2: Ground state error between real and calculated ground states.
• Figure 3: Energy error and optimal step sizes of schemes.
• Figure 4: Wave function error and corresponding error ratio.

Theorems & Definitions (30)

• Lemma 2.2
• Lemma 2.3: norm equivalence
• Theorem 3.1: energy dissipation for the $H_0^1$ scheme
• Theorem 3.2: energy dissipation for the $a_0$ scheme
• Remark 3.3
• Theorem 3.4: global convergence for the $H_0^1$ scheme
• Theorem 3.5: global convergence for the $a_0$ scheme
• Definition 3.6
• Lemma 3.8
• Theorem 3.9: local linear convergence for the $H_0^1$ scheme