Wei–Norman factorization for <i>S</i>(<i>U</i>(1) × <i>U</i>(1, 1))

In this paper, we concentrate on a class of nonlinear and non-autonomous first-order differential equations obtained through the Wei–Norman factorization method on the Lie group S(U(1) × U(1, 1)). These equations have certain applications in quantum control. By the Pontryagin maximum principle, we consider the explicit optimal control functions on the Lie algebra of S(U(1) × U(1, 1)). Together with two-stage Gauss–Legendre approach and Newton iterative method, we study the numerical solutions of nonlinear and non-autonomous first-order differential equations. We conduct numerical experiments to verify our results. The global error analysis indicates that this method is effective and practical.