LAUSR

A (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation with variable coefficients is considered, and two kinds of reduction are presented. Based on these transformations, via Darboux transformation method and Hirota method, nonlocal and localized spatiotemporal soliton solutions are constructed. In the first kind of reduction, variables x and y are mixed in the single formal variable X; thus, we cannot construct completely localized structures in x and y directions. In order to discuss completely localized structures, we consider the second kind of reduction, where variables x and y are independently included in two formal variables X and Y, respectively. Based on two kinds of reduction and the related solutions, nonlocal and localized spatiotemporal soliton structures are investigated.