Three-dimensional exact solutions of Gross–Pitaevskii equation with variable coefficients

A three-dimensional Gross–Pitaevskii (3D-GP) equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. We study exact solutions of the 3D-GP equation with (space, time)-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of exact solutions of the 3D-GP equation with different amplitude surfaces, in which some physically relevant solutions are described. These results may raise the possibility of relative experiments and potential applications.