Bright and dark optical vortex solitons of (3Â +Â 1)-dimensional spatially modulated quintic nonlinear Schrödinger equation

We investigate a (3 + 1)-dimensional spatially modulated quintic nonlinear Schrödinger equation with external potential, and obtain bright and dark vortex soliton solutions. Based on these analytical solutions, bright and dark vortex soliton structures are constructed. Our analysis and results help the comprehension of vortex solitons in nonlinear optics and Bose-Einstein condensation.