We consider the computations of the action ground state for a rotating nonlinear Schr\"odinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

翻译：我们考虑旋转非线性薛定谔方程的作用基态计算问题。该问题归结为在Nehari约束下最小化作用泛函。在聚焦情形中，我们找到了问题的等价表述形式，从而简化了约束条件。基于此，我们提出了一种带有渐近拉格朗日乘子的归一化梯度流方法，并建立了能量衰减性质。此外，还应用了流行的优化方法以提高效率。在散焦情形中，我们证明了基态可通过无约束最小化获得，随后应用了直接梯度流法和无约束优化方法。数值实验展示了所提方法在两种情形下的收敛性和精度，并讨论了效率对比。最后，对作用基态与能量基态之间的关系进行了数值研究。