Abstract

Anderson localization—so named after the American physicist P. W. Anderson—is the absence of diffusion of waves in a disordered medium. It is a generic wave phenomenon, which applies to any kind of wave regardless of its nature (light, acoustic, matter, spin, etc.). The localization occurs because strong disorder induces multiple scattering paths along which the components of the wave field interfere destructively. Given this insight, it has been discussed that a small (Ginzburg-Landau style) nonlinearity might destroy the localized state (because overlapping components of the nonlinear field produce internal pressure favoring spreading). The phenomenon has been discussed theoretically and demonstrated numerically suggesting that above a certain critical strength of nonlinear interaction a (weakly) nonlinear field may propagate to infinite distances despite the underlying disorder—in contrast to the corresponding linear field. The statistics of this spreading process, as well as the type of nonlinearity destroying localization, has remained a matter of debate. In this talk, I will review the state of the art, with several toy models predicting asymptotic spreading of the wave field from the discrete nonlinear Schrödinger equation with random potential. The keywords will be continuous time random walks, chaos (strong, weak), percolation, fractional kinetics, Cayley trees. Time permitting, I will touch upon topics concerning the nonlinear Schrödinger models with subquadratic power nonlinearity. The main issue here is that subquadratic power represents a form of long-range self-interaction in the system and as such might have important implications with regard to the asymptotic spreading process. A summary of the talk can be found in a recent publication [A.V. Milovanov and A. Iomin, Phys. Rev. E. 107, 034203 (2023)].