Higher-dimensional soliton structures of a variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential

reduction correlation between the (3+1) -dimensional variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential and a (2+1) -dimensional constant-coefficient one is firstly erected. With the aid of solutions via the Hirota method for the (2+1) -dimensional constant-coefficient equation, the (3+1) -dimensional soliton analytical solutions with the Hermite–Gaussian envelope including vortex, diploe soliton and saddle-shaped soliton are firstly unfolded. Expanded and compressed evolutions of these (3+1) -dimensional soliton structures are presented in the periodic amplification and exponential diffraction decreasing systems. In the x-z plane, the eye-shaped structure appears in all soliton structures, and its number is related to the Hermite parameter.