论文标题
在非线性量子图上计算固定态的梯度流量
Gradient Flow Approach to the Calculation of Stationary States on Nonlinear Quantum Graphs
论文作者
论文摘要
我们介绍并实施了一种计算公制图上非线性schr \'odinger方程的固定态的方法。固定状态作为固定质量下的非线性Schr \''Odinger能量的局部最小化。我们的方法基于能量的归一化梯度流（即投射在固定质量球上的梯度流），适用于非线性量子图的上下文。我们首先证明，在连续水平上，归一化梯度流量良好，质量保护，能量减少和收敛（至少在局部）向固定状态。然后，我们建立连续流与其离散版本之间的链接。在模型情况下，我们在模型情况下进行了一系列数值实验，以表明离散流动以计算固定状态的良好性能。同伴论文中给出了进一步的实验以及我们数值算法的详细说明。
We introduce and implement a method to compute stationary states of nonlinear Schr\''odinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schr\''odinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.
