Vortex soliton solutions of a (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential

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.