A (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential is followed with interest. The mapping procedure from the distributed-coefficient to the constant-coefficient equations is provided. Via the procedure with solution of constant-coefficient NLSE from the variational principle, an approximate vortex soliton solution is reported. With the increase of value for topological charge m, torus-shaped vortex soliton turns into ring vortex soliton, and central region of ring gradually enlarges. The layer of vortex solitons and their phase along the z-direction is related to the number of \(q+1\) with the Hermite parameter q. The branch number of spiral phase structures for vortex solitons is related to the value of topological charge m. The stability test from the direct numerical simulation indicates that vortex solitons with \(m=1,2,3,\, q=0\) are stable, otherwise, vortex solitons with other values of m, q are unstable.

感兴趣的是一个（3 +1）维Gross-Pitaevskii方程，它在线性势下具有部分非局部分布的系数。提供了从分布系数到常数系数方程的映射过程。通过变分原理，采用常系数NLSE的求解方法，得到了近似的涡旋孤子解。随着拓扑电荷m值的增加，圆环形涡旋孤子变成环形涡旋孤子，并且环的中心区域逐渐增大。涡旋孤子的层及其沿z方向的相位与带有Hermite参数q的\（q + 1 \）的数量有关。涡旋孤子的螺旋相结构的分支数与拓扑电荷m的值有关。直接数值模拟的稳定性测试表明\（m = 1,2,3，\，q = 0 \）的涡旋孤子是稳定的，否则，m，q的其他值的涡旋孤子是不稳定的。