In this work, we study some mathematical features for the action ground states of the defocusing nonlinear Schr\"odinger equation with possible rotation. Main attention is paid to characterizing the relation between the action ground states and the energy ground states. Theoretical equivalence and non-equivalence results have been established. Asymptotic behaviours of the physical quantities are derived in some limiting parameter regimes. Numerical evidence of non-equivalence is observed and numerical explorations for vortices phenomena in action ground states are done.

翻译：本文研究了带有可能旋转的散焦非线性薛定谔方程的作用基态的一些数学性质。主要关注点在于刻画作用基态与能量基态之间的关系。我们建立了理论上的等价性与非等价性结果。在若干极限参数区域中，推导了物理量的渐近行为。观察到了非等价性的数值证据，并对作用基态中的涡旋现象进行了数值探索。