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.

Figures (7)

• Figure 5.1: Radial 3D ground-state profiles for different masses $c=1,4,8,12$ with $\beta=-10$ and $\lambda=0.1$. The left panel shows the free-space case, and the right panel shows the harmonic-potential case $V(r)=10^5r^2$.
• Figure 5.2: Radial 3D ground-state profiles for $\beta=-20,-30,-40,-50$ with $\lambda=4$. The left panel shows the free-space case, and the right panel shows the harmonic-potential case $V(r)=10^5r^2$.
• Figure 5.3: Radial 3D ground-state profiles for $\lambda=0.25,0.5,0.75,1$ with fixed $\beta=-10$. The left panel shows the free-space case, and the right panel shows the harmonic-potential case $V(r)=10^5r^2$.
• Figure 5.4: Heatmap of $\eta_\theta$ in the $(1/\lambda,|\beta|)$-plane for $-25<\beta<-1$ and $1/500<\lambda<1$, showing the droplet-like, soliton-like, and no-ground-state regimes. The dashed curve denotes the contour line corresponding to the threshold $\eta_\theta=0.62$ used to separate the droplet-like and soliton-like regions. Two representative radial ground-state profiles corresponding to the marked points $A$ and $B$ are displayed on the right.
• Figure 5.5: Two-dimensional computation without an external potential. The left panel shows the ground-state density, while the right panel shows the corresponding adaptive mesh.

Theorems & Definitions (27)

• Remark 2.1
• Theorem 3.1: Free-space case
• Theorem 3.2: Confining potential case
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof