A model for cold dark matter is given by the solution of a coupled Schrödinger–Poisson equation system. We present a numerical scheme for integrating these equations, discussing the problems arising from their nonlinear and nonlocal character. After introducing and testing our numerical approach, we illustrate key features of the system by numerical examples in 1 + 1 dimensions. In particular, we study the properties of asymptotic states to which the numerical solutions converge for artificial initial conditions.