(1+1)-Dimensional Schr\"odinger-Poisson equation with contact interaction
We investigate the role of contact interactions in the dynamics of fuzzy dark matter (FDM) modeled through the Schr\"odinger-Poisson equation in one spatial dimension. While the $\Lambda$CDM paradigm successfully explains structure formation on large scales, its small-scale predictions remain in tension with observations. FDM offers an alternative framework, where local self-interactions can further influence the formation and evolution of structures. We explore both attractive and repulsive contact interactions in static and expanding backgrounds. Using numerical simulations, we examine their impact on three key scenarios: the properties of the lowest-energy stationary solution, the relaxation of localized initial states, and the gravitational collapse of nonlocalized states. Our results show that contact interactions modify the density profile of the stationary solution and affect the onset of characteristic stages of gravitational collapse, particularly the shell-crossing event. In the (1+1) model, we confirm that relaxation does not converge to the lowest-energy stationary solution, even when local self-interactions are included. Taken together, local self-interactions play a relevant role in shaping the nonlinear dynamics of FDM and motivate further studies in higher-dimensional and cosmologically realistic settings.