Two-stage continuation algorithms for Bloch waves of Bose–Einstein condensates in optical lattices

Bloch waves of Bose–Einstein condensates (BEC) in optical lattices are extremum nonlinear eigenstates which satisfy the time-independent Gross–Pitaevskii equation (GPE). We describe an efficient Taylor predictor–Newton corrector continuation algorithm for tracing solution curves of parameter-dependent problems. Based on this algorithm, a novel two-stage continuation algorithm is developed for computing Bloch waves of 1D and 2D Bose–Einstein condensates (BEC) in optical lattices. We split the complex wave function into the sum of its real and imaginary parts. The original GPE becomes a couple of two nonlinear eigenvalue problems defined in the real domain with periodic boundary conditions. At the first stage we use the chemical potential μ as the continuation parameter. The Bloch wavenumber k  (kx,ky)(kx,ky), and the coefficient of the cubic term are treated as the second and third continuation parameters, respectively. Then we compute the Bloch bands/surfaces for the 1D/2D problem with linear counterparts. At the second stage we use μ   and k/kxk/kx or kyky as the continuation parameters simultaneously with two constraint conditions. The states without linear counterparts in the GPE can be obtained via states with linear counterparts. Numerical results are reported for both 1D and 2D problems.