From optics to dark matter: A review on nonlinear Schrödinger–Poisson systems

Abstract

We review recent progress in Schrödinger–Poisson systems in 1+2 and 1+3 dimensions in the presence of nonlinear terms. In a mean field approach, this mathematical model describes the semiclassical behavior of an -body system of identical bosons with nonlocal interactions between them. The 1+2D model can be used to describe the nonlinear propagation of optical beams in thermo-optical media and can be regarded as an analog photonic system for a self-gravitating self-interacting wave, which is the situation of the full 1+3D case, representing the dynamics of coherent dark matter under the assumption that it is made up of ultralight axions. After providing a rough overview of the disparate physical contexts in which the Schrödinger–Poisson equation has been applied, we discuss the main ideas and a number of recent findings in the two aforementioned frameworks. For both setups, we present families of stationary solutions, including vortex states, and discuss the implications of the simulation of propagation dynamics in a number of cases of interest. Finally, we discuss some numerical methods to solve the system of time-dependent partial differential equations.