Hermite–Gaussian vortex solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrödinger equation with variable coefficients

We consider the wave motion in a partially nonlocal and inhomogeneous nonlinear medium, and a (3+1)-dimensional nonlocal nonlinear Schrodinger equation with variable coefficients is used to govern this dynamics. Based on this model, spatiotemporal Hermite–Gaussian vortex soliton solutions are derived. The evolution behaviors of spatiotemporal Hermite–Gaussian vortex solitons in a diffraction decreasing system are investigated. Results indicate that the topological charge m changes the spiral structures of phase, and its value determines the number of the branch of the spiral phase structures. If the value of parameter n adds, spatiotemporal vortex solitons change their structures. Obviously, the layer of ring solitons along the vertical (z-axis) direction is decided by \(n+1\).