On action ground states of defocusing nonlinear Schrödinger equations

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.