A review of dispersive limits of (non)linear schrödinger-type equations

In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrödinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schrödinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schrödinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.