The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross-Pitaevskii equations with nonzero boundary conditions in dimension one has been established recently in [de Laire and S. L\'opez-Mart\'inez, Comm. Partial Differential Equations, 2022]. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. This paper provides a numerical method to compute approximations of such solitons for these types of equations, and provides actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow us to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow us to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.

翻译：非局部非线性Schrödinger方程的暗孤子的数值计算 最近，在一维非零边界条件下的一大类非线性非局部Gross-Pitaevskii方程的存在性和衰减性质已经在[ de Laire and S. L\'opez-Mart\'inez，Comm. Partial Differential Equations, 2022]中得到确定。数学上，这些孤子对应于在固定动量下的能量的极小值，并且是轨道稳定的。本文提供了一种计算这些方程的此类孤子的近似值的数值方法，并为几种具有物理相关的非局部势提供了实际的数值实验。这些模拟使我们能够获得各种黑暗的孤子，并根据非局部势的参数评论它们的形状。特别地，它们表明，给定色散关系，声速和朗道速度是理解这些暗孤子的特性的重要值。它们还允许我们测试证明暗孤子存在性的一些充分条件的必要性。