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

In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrodinger type equations. Different f ormulations are used to study these singular limits, e.g., the kinetic form ulation of the linear Schrodinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) tech niques, while the modified Madelung transformation reformulating Schrodinge r equations in terms of a dispersive perturbation of a quasilinear symmetri c hyperbolic system usually only gives local-in-time results due to the hyp erbolic nature of the limit equations. Deterministic analogues of turbulenc e are also discussed. There, turbulent diffusion appears naturally in the z ero dispersion limit.