A G-FDTD scheme for solving multi-dimensional open dissipative Gross-Pitaevskii equations

Behaviors of dark soliton propagation, collision, and vortex formation in the context of a non-equilibrium condensate are interesting to study. This can be achieved by solving open dissipative Gross-Pitaevskii equations (dGPEs) in multiple dimensions, which are a generalization of the standard Gross-Pitaevskii equation that includes effects of the condensate gain and loss. In this article, we present a generalized finite-difference time-domain (G-FDTD) scheme, which is explicit, stable, and permits an accurate solution with simple computation, for solving the multi-dimensional dGPE. The scheme is tested by solving a steady state problem in the non-equilibrium condensate. Moreover, it is shown that the stability condition for the scheme offers a more relaxed time step restriction than the popular pseudo-spectral method. The G-FDTD scheme is then employed to simulate the dark soliton propagation, collision, and the formation of vortex-antivortex pairs.