We present an analysis of stationary solutions for two-dimensional (2D) Bose–Einstein condensates (BECs) with the Rashba spin–orbit (SO) coupling and Zeeman splitting. By introducing the generalized momentum operator, the linear version of the system can be solved exactly. The solutions are semi-vortices of the Bessel-vortex (BV) and modified Bessel-vortex (MBV) types, in the presence of the weak and strong Zeeman splitting, respectively. The ground states (GSs) of the full nonlinear system are constructed with the help of a specially designed neural network (NN). The GS of the mixed-mode type appears as cross-attraction interaction increases. The spin texture of the GS is produced in detail. It exhibits the Néel skyrmion structure for the semi-vortex GS of the BV type, and the respective skyrmion number is found in an analytical form. On the other hand, GSs of the MBV and mixed-mode types do not form skyrmions.

我们提出了对具有 Rashba 自旋轨道 (SO) 耦合和塞曼分裂的二维 (2D) 玻色-爱因斯坦凝聚体 (BEC) 的固定解的分析。通过引入广义动量算子，可以精确求解系统的线性版本。解是贝塞尔涡旋 (BV) 和修正贝塞尔涡旋 (MBV) 类型的半涡旋，分别存在弱塞曼分裂和强塞曼分裂。整个非线性系统的基态 (GS) 是在专门设计的神经网络 (NN) 的帮助下构建的。混合模式类型的 GS 随着交叉吸引力相互作用的增加而出现。GS的自旋纹理被详细制作。它展示了 BV 型半涡旋 GS 的 Néel skyrmion 结构，并以解析形式找到了相应的 skyrmion 数。