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：对于一类具有非零边界条件的一维非局域Gross-Pitaevskii方程，其暗孤子的存在性与衰减性质最近在[de Laire和S. L\'opez-Mart\'inez, Comm. Partial Differential Equations, 2022]中得到了严格证明。从数学角度看，这些孤子对应固定动量下能量的极小值，并且具有轨道稳定性。本文针对这类方程提出了一种计算暗孤子近似解的数值方法，并对多种具有物理相关性的非局域势进行了实际数值实验。这些模拟使我们能够获得不同类型的暗孤子，并基于非局域势的参数对其形状进行评述。特别地，数值结果表明，给定色散关系后，声速与朗道速度是理解这些暗孤子性质的重要参数。这些实验还使我们能够检验证明暗孤子存在性的理论结果中某些充分条件的必要性。