Mathematical and numerical studies on ground states of the extended Gross-Pitaevskii equation with the Lee-Huang-Yang correction

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.

• The paper establishes precise existence and nonexistence regimes for ground states, highlighting a critical mass threshold for attractive interactions in 2D and 3D.
• It employs variational methods with dimensional reduction and finite element simulations to capture transitions from soliton-like to droplet-like states.
• A robust normalized gradient flow using Lagrange multipliers is introduced, ensuring stability and convergence for high-dimensional quantum droplet computations.

Mathematical and Numerical Analysis of Ground States in the Extended Gross–Pitaevskii Equation with Lee–Huang–Yang Correction

Introduction and Context

The paper rigorously addresses the study of ground states in the extended Gross–Pitaevskii (eGP) equation that incorporates the Lee–Huang–Yang (LHY) correction, a higher-order quantum fluctuation term essential for accurately modeling quantum droplets in ultracold Bose gases. The eGP model generalizes the classical Gross–Pitaevskii equation by including a quintic nonlinearity derived from the seminal work of Lee, Huang, and Yang on beyond mean-field effects. The emergence of self-bound quantum droplets, first predicted by Petrov and verified experimentally, motivates the mathematical and algorithmic investigation of ground state properties in systems where a delicate balance between attractive and repulsive contributions controls localization phenomena.

Model Formulation and Dimensional Reduction

The eGP equation considered can be written as:

$$i\partial_t \psi(\mathbf{x}, t) = \left[ -\frac{1}{2} \nabla^2 + V(\mathbf{x}) + \beta |\psi|^2 + \lambda |\psi|^3 \right]\psi,$$

where $\beta$ is the mean-field (cubic) interaction strength and $\lambda > 0$ quantifies the LHY correction in the effective nonlinear Schrödinger description. The system conserves the mass $\|\psi\|_2 = c$ and is subject to both free-space and confining external potentials.

Through systematic nondimensionalization, the authors derive consistent reduced models in 1D, 2D, and 3D via rigorous dimensional reduction. In the presence of strong harmonic confinement, the full three-dimensional problem is reduced to lower-dimensional eGP equations by integrating out tightly confined directions, leading to modified effective coefficients that encode the ground mode structure of the frozen degrees of freedom.

Existence and Nonexistence Theory for Ground States

The main analytical contribution establishes comprehensive regimes for the existence and nonexistence of ground states under the mass constraint in various dimensions, both for free space and for confining potentials:

• Free-Space Case: With $V \equiv 0$, the existence of ground states crucially depends on the sign of $\beta$ and the value of the mass $c$:
  • For $\beta \geq 0$: No ground state exists for any $c$; the energy infimum is always $0$ and unachievable.
  • For $\beta$0 and $\beta$1: Minimizers exist for all masses.
  • For $\beta$2 and $\beta$3: There is a sharp mass threshold $\beta$4, explicitly characterized by variational methods, below which no minimizer exists, and above which the ground state exists.
• Confining Potential: For external potentials $\beta$5 diverging at infinity, existence is guaranteed for all interaction parameters and masses.

These results are derived via direct energy minimization under $\beta$6-normalization, employing rearrangement inequalities, compactness theorems, and sharp interpolation estimates to treat the competing cubic-quintic structure. The monotonicity properties with respect to mass and the sign transitions in the energy infimum as a function of the system parameters are made explicit.

Numerical Schemes: Normalized Gradient Flow with Lagrange Multipliers

For computational construction of ground states, the paper extends the normalized gradient flow with discrete normalization, leveraging Lagrange multipliers for the mass constraint and implicit-explicit (IMEX) time discretizations tailored for stability in the presence of the high-degree nonlinearity. At each iteration, a linear elliptic problem is solved for the intermediate state, followed by normalization. The scheme guarantees preservation of nonnegativity and robust convergence properties.

Spatial discretization employs finite element methods on general domains, facilitating adaptive mesh refinement crucial for resolving sharply localized states typical of self-bound quantum droplets. For cases with high symmetry, the authors utilize radial reductions and efficient finite-difference implementations.

Numerical Results and Regime Classification

A thorough suite of simulations is presented:

• Parameter Dependence: Systematic variations in mass $\beta$7, cubic interaction $\beta$8, and LHY strength $\beta$9 reveal how ground-state profiles transition from smooth, soliton-like decay to flat-top, droplet-like structures as attraction increases or quantum pressure decreases.
• Phase Diagram: The $\lambda > 0$0 plane is mapped, exhibiting a tripartition into (i) no-ground-state, (ii) soliton-like, and (iii) droplet-like regimes, using a robust diagnostic measuring mass localization near the density maximum.
• Heuristic Flat-Top Approximation: In the strong-attraction or weak-LHY limit, the ground state is accurately approximated by a compactly supported, nearly constant profile. Analytical expressions for the plateau height and leading-order energy are derived and validated by exact numerical solutions, with relative errors vanishing as $\lambda > 0$1 or $\lambda > 0$2.
• High-Dimensional and External Potential Effects: The finite element algorithm is benchmarked in full 2D and 3D, including cases with optical lattice and anisotropic harmonic trapping. The method adaptively resolves interfaces and interfaces between localized droplets and background vacuum.

Implications and Future Directions

The analytical framework rigorously clarifies the critical behavior of the eGP ground states as system parameters vary and establishes that the LHY correction induces a nontrivial structure in the variational landscape absent in the standard cubic Gross–Pitaevskii theory. The unification of existence results in all dimensions, including the explicit mass thresholds for ground-state formation in free space with attractive interaction, provides insight into the parameter control required to realize quantum droplets or prevent collapse.

Numerically, the normalized gradient flow/fem approach extends to higher dimensions and enables systematic exploration of the parameter space and external potentials that more closely model realistic condensate experiments.

Open directions include extension to dipolar and multi-component systems, where nonlocal and intercomponent interactions further enrich the variational structure, and investigation of excited and metastable states. Dynamical stability analysis and real-time quantum droplet dynamics are also natural continuations given the framework.

Conclusion

This work provides a mathematically rigorous and algorithmically robust examination of the ground states in the extended Gross–Pitaevskii equation with the LHY correction. By systematically analyzing existence and nonexistence regimes and developing specialized numerical solvers, the paper advances both the theoretical understanding and practical computation of quantum droplet states in ultracold Bose gases, laying a foundation for future studies in complex many-body and anisotropic systems.

Reference: "Mathematical and numerical studies on ground states of the extended Gross-Pitaevskii equation with the Lee-Huang-Yang correction" (2604.04460).