A structure-preserving, second-order-in-time scheme for the von Neumann equation with power nonlinearity

In this paper we propose a structure-preserving, linearly implicit, second-order-in-time scheme for the numerical solution of the von Neumann equation with power nonlinearity (also known as the Alber equation). Fourth order finite differences are used for the spatial discretization. We highlight the importance of the correct initialization of the method in achieving the expected order of convergence in space and time. As illustrative examples, we investigate the bifurcation from Landau damping to modulation instability. In that context, amplification factors in the fully developed modulation instability for this nonlinear equation are computed for the first time.