On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an H⁴-regular solution, a first-order error bound in the H¹ norm is shown and used to derive a second-order error bound in the L₂ norm. For the cubic Schrödinger equation with an H⁴-regular solution, first-order convergence in the H² norm is used to obtain second-order convergence in the L₂ norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and Hᵐ-conditional stability for error propagation, where $m=1$ for the Schrödinger-Poisson system and $m=2$ for the cubic Schrödinger equation.