We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in Leung et al. (Methods Appl Anal 21(1):31–66, 2014) for the linear Schrödinger equation in the semi-classical regime to the nonlinear case with nonlocal potentials. To apply the methodology of FHSM effectively, we propose two schemes by using the Lie’s and the Strang’s operator splitting, respectively, so that one can handle the nonlinear nonlocal interaction term using the fast Fourier transform. The resulting algorithm can then enjoy the same computational complexity as in the linear case. Numerical examples demonstrate that the two operator splitting schemes achieve the expected first-order and second-order accuracy, respectively. We will also give one-, two- and three-dimensional examples to demonstrate the efficiency of the proposed algorithm.

我们提出了求解一类涉及非局部势能的非线性薛定谔方程的有效数值方法。这种非局部电位受粒子间非局部相互作用强度建模的高斯卷积控制。该方法扩展了我们在 Leung 等人中开发的快速惠更斯扫描方法 (FHSM)。（Methods Appl Anal 21(1):31–66, 2014）将半经典状态中的线性薛定谔方程用于具有非局部电位的非线性情况。为了有效地应用 FHSM 的方法，我们分别使用 Lie 和 Strang 算子分裂提出了两种方案，以便使用快速傅立叶变换处理非线性非局部相互作用项。结果算法可以享受与线性情况相同的计算复杂度。数值例子表明，两种算子分裂方案分别达到了预期的一阶和二阶精度。我们还将给出一维、二维和三维示例来证明所提出算法的效率。