This paper presents a numerical method for computing the ground states of the three‑dimensional Gross-Pitaevskii equation （GPE） that includes the Lee-Huang-Yang （LHY） correction.The LHY term in the Gross-Pitaevskii energy functional is of the form |ψ|5 <?fx-imagestate width="4.40266657" height="3.89466691"?> <?fx-imagestate width="4.40266657" height="3.89466691"?> and accounts for quantum fluctuations in Bose-Einstein condensates （BECs）.First， the continuous projected gradient flow is discretized with a Lagrange multiplier by using a Fourier pseudo-spectral method in space and a backward Euler scheme in time， obtaining a linearly implicit backward Euler scheme （PGF-BESP）.The key idea is to implicitly determine the Lagrange multiplier through an orthogonal condition， thereby guaranteeing mass conservation （the normalization constraint） and providing an explicit expression for the Lagrange multiplier directly from the orthogonal condition.This preserves the two fundamental properties of the continuous normalized gradient flow with a Lagrange multiplier： Mass conservation and energy dissipation.Next， we carry out a rigorous theoretical analysis of the scheme and prove that the discrete energy is monotonically non-increasing under a suitable condition on the time step size.We then perform numerical experiments to verify the theoretical results， and the results show that mass is indeed conserved and the energy decays monotonically.Finally， we use the proposed method to compute ground state configurations for different potential terms and different interaction parameters.We find that as the interaction parameter increases， the overall size of the ground state density distribution also increases， and under the optical lattice potential， multi-layer structures are observed.In summary， both theoretical and numerical results show that the proposed PGF-BESP method provides a stable， efficient and easy to implement numerical tool for ground state computations of GPE with LHY correction.
This paper presents a numerical method for computing the ground states of the three‑dimensional Gross-Pitaevskii equation （GPE） that includes the Lee-Huang-Yang （LHY） correction.The LHY term in the Gross-Pitaevskii energy functional is of the form |ψ|5 <?fx-imagestate width="4.40266657" height="3.89466691"?> <?fx-imagestate width="4.40266657" height="3.89466691"?> and accounts for quantum fluctuations in Bose-Einstein condensates （BECs）.First， the continuous projected gradient flow is discretized with a Lagrange multiplier by using a Fourier pseudo-spectral method in space and a backward Euler scheme in time， obtaining a linearly implicit backward Euler scheme （PGF-BESP）.The key idea is to implicitly determine the Lagrange multiplier through an orthogonal condition， thereby guaranteeing mass conservation （the normalization constraint） and providing an explicit expression for the Lagrange multiplier directly from the orthogonal condition.This preserves the two fundamental properties of the continuous normalized gradient flow with a Lagrange multiplier： Mass conservation and energy dissipation.Next， we carry out a rigorous theoretical analysis of the scheme and prove that the discrete energy is monotonically non-increasing under a suitable condition on the time step size.We then perform numerical experiments to verify the theoretical results， and the results show that mass is indeed conserved and the energy decays monotonically.Finally， we use the proposed method to compute ground state configurations for different potential terms and different interaction parameters.We find that as the interaction parameter increases， the overall size of the ground state density distribution also increases， and under the optical lattice potential， multi-layer structures are observed.In summary， both theoretical and numerical results show that the proposed PGF-BESP method provides a stable， efficient and easy to implement numerical tool for ground state computations of GPE with LHY correction.
The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this paper, we revisit the time discretizations for the GFDN and its generalization to the multicomponent BECs. Several widely used time discretizations are demonstrated to be not accurate for computing the ground state solution in the general case, especially for the multicomponent BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow. Through analysis and numerical computation, we clarify that, in order to accurately compute the ground state solution, the GFDN method must be discretized in very special ways in time such as the linearized backward Euler scheme, while the GFLM can be discretized by various schemes and works for multicomponent BEC with multiple constraints. We propose a simple backward-forward Euler scheme to discretize the GFLM, where only a linear system with constant coefficients needs to be solved at each time step. Taking the spin-1 BEC as an example, we demonstrate how the traditional GFDN approach fails in computing the ground state solution of general multicomponent BECs accurately. The proposed GFLM with backward-forward Euler discretization is shown to be accurate and efficient. Extensive numerical results are provided to verify our claims.
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.
In this paper, we present a systematic study on the ground state computation of quantum droplets in homonuclear Bose-Bose mixtures, governed by the extended Gross-Pitaevskii equations (eGPEs) with Lee-Huang-Yang (LHY) corrections. This model captures the formation of self-bound droplets stabilized by the delicate balance between the attractive mean-field interaction and the repulsive quantum fluctuations. We formulate dimensionless energy functionals for both the general two-component system and the reduced single-component density-locked model. To compute the ground states efficiently, we adapt and benchmark various gradient flow discretization schemes, identifying a backward-forward sine-pseudospectral scheme based on the gradient flow with Lagrange multiplier method (GFLM-BFSP) as the robust solver for our simulations. Utilizing this method, we report three main numerical observations: (i) the density-locked model is quantitatively validated as a reliable approximation for ground state properties; (ii) the dimension-dependent convergence rates of the Thomas-Fermi approximation are established in the strong-coupling regime; and (iii) the critical particle number for self-binding in free space is numerically determined, providing a precise correction to the analytical prediction by Petrov [Phys. Rev. Lett. 115, 155302 (2015)].
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 have 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 for 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.
In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose–Einstein condensates (BECs) by using the normalized gradient flow or imaginary time method. The key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization. This allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of a single-component BEC to compute the ground state of spin-1 BECs. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method, is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BECs with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.
The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. We apply GFSI to the two-component scenario with Josephson junction and rotating term, which is one of the most important and topical models in multi-component Bose-Einstein condensates (MBECs), and rigorously establish the following fundamental results for the first time. By introducing a Lagrange multiplier to reformulate GFSI into an equivalent form, we prove its energy dissipation property and global convergence to stationary states. More significantly, we uncover an intrinsic connection between this classical numerical PDE discretization rooted in imaginary-time evolution and Riemannian optimization, a state-of-the-art mathematical framework for manifold-constrained optimization. This connection enables us to fully characterize the local convergence behavior of GFSI within the Riemannian optimization framework. Together with the aforementioned global convergence result, this yields a complete global--local convergence theory for GFSI. Finally, numerical experiments comprehensively validate the theoretically predicted energy dissipation and convergence properties.