  We present two accelerated numerical algorithms for single-component and
binary Gross-Pitaevskii (GP) equations coupled with microwaves (electromagnetic
fields) in steady state. One is based on a normalized gradient flow
formulation, called the ASGF method, while the other on a perturbed, projected
conjugate gradient approach for the nonlinear constrained optimization, called
the PPNCG method. The coupled GP equations are nonlocal in space, describing
pseudo-spinor Bose-Einstein condensates (BECs) interacting with an
electromagnetic field. Our interest in this study is to develop efficient,
iterative numerical methods for steady symmetric and central vortex states of
the nonlocal GP equation systems. In the algorithms, the GP equations are
discretized by a Legendre-Galerkin spectral method in a polar coordinate in
two-dimensional (2D) space. The new algorithms are shown to outperform the
existing ones through a host of benchmark examples, among which the PPNCG
method performs the best. Additional numerical simulations of the central
vortex states are provided to demonstrate the usefulness and efficiency of the
new algorithms.
