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：最近，[de Laire和S. L'ópez-Martínez, 偏微分方程通讯, 2022] 针对一大类具有非零边界条件的非线性非局域Gross-Pitaevskii方程，在维度为一的情况下建立了暗孤子的存在性与衰减性质。从数学角度看，这些孤子对应于固定动量下能量的极小化子，并具有轨道稳定性。本文提出了一种数值方法，用于计算这类方程中此类孤子的近似解，并针对数种具有物理意义的非局域势提供了实际的数值实验。这些模拟使我们能够获得多种暗孤子，并针对非局域势的参数对其形态进行讨论。特别地，模拟结果表明，给定色散关系，声速和朗道速度是理解这些暗孤子性质的重要数值。它们也使我们能够检验证明暗孤子存在的理论结果中某些充分条件的必要性。