Efficient and accurate methods for solving the time-dependent spin-1 Gross-Pitaevskii equation

We develop a numerical method for solving the spin-1 Gross-Pitaevskii equation. The basis of our work is a two-way splitting of the spin-1 evolution equation that leads to two exactly solvable flows. We use this to implement a second-order and a fourth-order symplectic integration method. These are the first fully symplectic methods for evolving spin-1 condensates. We develop two non-trivial numerical tests to compare our methods against two other approaches.