Analytical spatiotemporal similaritons for the generalized (3 + 1)-dimensional Gross–Pitaevskii equation with an external harmonic trap

We study the generalized (3 + 1)-dimensional Gross–Pitaevskii equation with time-dependent nonlinearity, harmonic trap and gain or loss, and obtain bright and dark spatiotemporal similaritons, whose form originates from the exact balance between the nonlinearity, harmonic trap and the gain/loss. Based on these analytical solutions, we firstly discuss the solitonic similaritons interaction and control in a exponential amplification system. Secondly, we investigate the nonlinear tunnelling of bright and dark similariton pairs passing through the nonlinear barrier and well. Results indicate that the similariton can be compressed to a desired width and amplitude in a controllable manner by the choice of the barrier (or well) parameters.

► Spatiotemporal similaritons of the generalized 3D GP equation.
► Solitonic similaritons interaction and control.
► Tunneling effects of bright and dark similaritons.