论文标题
通过归一化梯度流量计算非线性schrödinger方程的最小动作基态
Computing the least action ground state of the nonlinear Schrödinger equation by a normalized gradient flow
论文作者
论文摘要
在本文中，我们概括了归一化梯度流量方法，该方法首先应用于计算最少的能量基态以计算最小动作基态。将提出连续的归一化梯度流（CNGF），并将作用降低的属性证明可以通过离散归一化（GFDN）提供梯度流量的数学合理性。然后，我们使用向后的Euler方法在及时进一步离散GFDN，从而导致GFDN-BF方案。结果表明，GFDN-BF方案保留了阳性并无条件地减少动作。我们将其与其他三个方案进行了比较，这些方案是根据针对最少能量基础状态设计的相应的方案进行了比较的，数值结果表明，GFDN-BF方案在准确性，效率和稳健性方面的性能要比其他方案要好得多。提供了几种类型电位的最小动作基态的广泛数值结果。我们还使用数值结果来验证一些现有结果并导致一些猜想。
In this paper, we generalize the normalized gradient flow method which was first applied to computing the least energy ground state to compute the least action ground state. A continuous normalized gradient flow (CNGF) will be presented and the action diminishing property will be proved to provide a mathematical justification of the gradient flow with discrete normalization (GFDN). Then we use backward-forward Euler method to further discretize the GFDN in time which leads to the GFDN-BF scheme. It is shown that the GFDN-BF scheme preserves the positivity and diminishes the action unconditionally. We compare it with other three schemes which are modified from corresponding ones designed for the least energy ground state and the numerical results show that the GFDN-BF scheme performs much better than the others in accuracy, efficiency and robustness for large time steps. Extensive numerical results of least action ground states for several types of potentials are provided. We also use our numerical results to verify some existing results and lead to some conjectures.
