A second-order-in-time scheme for the von Neumann equation with singular self-interaction and simulation of the onset of instability
The von Neumann equation with delta self-interaction kernel serves as a statistical model for nonlinear waves, and it exhibits a bifurcation between stable and unstable regimes. In oceanography it is known as the Alber equation, and its bifurcation is important for understanding rogue waves, a key problem in marine safety. Despite its significance, only one first-order-in-time numerical method exists in the literature. In this paper, we propose a structure-preserving, linearly implicit, second-order-in-time scheme for its numerical solution. We employ fourth-order finite differences for the spatial discretization. As an illustrative example, we explore the onset of modulation instability. We verify that the linear stability analysis accurately predicts the initial growth phase, but fails to forecast the maximum amplitude, the formation of a coherent structure in the nonlinear regime, or the relevant timescales. Monte Carlo simulations with Gaussian background spectra reveal that the maximum amplitude depends mainly on the homogeneous background rather than the initial inhomogeneity. For weak instabilities, the inhomogeneity grows substantially from its initial condition, but remains small compared to the background. On the other hand, strong instability leads to recurrent hotspots of increased variance. This provides a possible explanation of how modulation instability makes rogue waves more likely in unidirectional sea states.