We investigate the action ground states of the defocusing nonlinear Schr\"odinger 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.

翻译：本文研究了带旋转与不带旋转的散焦非线性薛定谔方程的作用基态。我们的主要目标是刻画作用基态与能量基态之间的关系。理论上，我们证明了在非旋转情形下两者完全等价，而在旋转情形下则存在条件等价关系。我们的理论结果得到了大量数值实验的支持。值得注意的是，在旋转情形中，我们提供了不等价的数值示例，表明不等价性通常发生在作用基态中涡旋数量增加时的转变点附近。此外，我们研究了在某些极限参数区域下作用基态及其相关物理量的渐近行为，数值结果验证并补充了我们的理论分析。进一步地，我们通过数值方法探索了作用基态中涡旋图案的形成与演化。