On action ground states of defocusing nonlinear Schrödinger equations

TL;DR

This work analyzes action ground states of the defocusing nonlinear Schrödinger equation in both non-rotating and rotating frames, establishing complete equivalence with energy ground states when $\Omega=0$ and proving conditional equivalence for $\Omega\neq0$ along with a precise characterization of non-equivalence regions. The authors develop a robust variational framework based on $S_{\Omega,\omega}$, the Nehari manifold, and an unconstrained reformulation, and they supplement theory with a dual formulation and detailed asymptotic analyses as $\omega$ approaches key spectral thresholds and in large- or small- $\omega$ regimes. Numerically, they reveal vortex formation and transitions, identify a critical angular velocity $\Omega^c$ governing vortex appearance, and demonstrate non-equivalence at vortex-transition points, thereby enriching the understanding of ground-state structure in rotating Bose–Einstein condensates and related NLS models. The results provide both rigorous insight into mass-action relationships and practical computational tools to explore vortex patterns and asymptotics in rotating defocusing NLS systems.

Abstract

We investigate the action ground states of the defocusing nonlinear Schrödinger equation with and without rotation. Our primary focus is on characterizing the relationship between the action ground states and the energy ground states. Theoretically, we prove a complete equivalence of the two in the non-rotating case and a conditional equivalence in the rotating case. Our theoretical results are supported by extensive numerical experiments. Notably, in the rotating case, we provide numerical examples of non-equivalence showing that non-equivalence typically occurs at the transition points where the number of vortices in the action ground state is increasing. Additionally, we study the asymptotic behaviour of the action ground states and the associated physical quantities in certain limiting parameter regimes, with numerical results validating and complementing our analysis. Furthermore, we explore the formation and change of the vortex pattern in the action ground states numerically.

Figures (12)

• Figure 5.1: Change of the action $S_g(\Omega,\omega)$ and the mass $M_g(\Omega,\omega)$ with respect to $\omega$. Left: $\Omega=0$; Right: $\Omega=0.5$.
• Figure 5.2: Asymptotic rates of $S_g(\Omega,\omega)$ and $M_g(\Omega,\omega)$ when $\omega\to-\lambda_0(\Omega)^-$. Left: $\Omega=0$; Right: $\Omega=0.5$.
• Figure 5.3: Asymptotic rates of $S_g(\Omega,\omega)$ and $M_g(\Omega,\omega)$ when $\omega\to -\infty$. Left: $\Omega=0$; Right: $\Omega=0.5$.
• Figure 5.4: The rescaled action ground state $\tilde{\phi}_\omega$ (\ref{['rescaled state']}) for different $\omega$ and the limit $\tilde{\phi}$.
• Figure 5.5: Density profiles of $|\phi_{\Omega,\omega}(x_1,x_2)|^2$ around the critical $\Omega^c$: $0.766<\Omega^c<0.767$ for $\omega=-2$ (left column), $0.292<\Omega^c<0.293$ for $\omega=-10$ (center column), $0.089<\Omega^c<0.090$ for $\omega=-50$ (right column), where the lower bounds give non-vortex states (1st row) and the upper bounds give vortex states (2nd row).

Theorems & Definitions (51)

• lemma 1: Compact embeddingBaoCai
• proposition 1: Unconstrained minimizationLYZ
• remark 1
• theorem 2
• proof
• theorem 3
• lemma 2: Uniqueness
• proof
• remark 2
• lemma 3