Abstract

Anderson localization is the absence of diffusion of waves in random media. It is a generic wave phenomenon, which applies to any kind of wave regardless of its nature. Experimentally, Anderson localization has been found for electron gases, acoustic waves, spin waves, matter waves, and more recently also for light waves. The localization occurs because a disordered medium induces multiple scattering paths along which the components of the wave function interfere destructively. Lately, it has been discussed that a weak nonlinearity might destroy the localized state giving rise to unlimited spreading of the wave function along the lattice despite the underlying disorder. The statistics of this spreading process has remained a matter of debate. In this talk, I will review the state of the art, with several toy models predicting asymptotic spreading from the nonlinear Schrödinger dynamics on a lattice. The key words will be continuous time random walks, chaos (strong, weak), percolation, fractional kinetics, Cayley trees. Time permitting, I will touch upon topics concerning nonlinear Schrödinger models with subquadratic power nonlinearity leading to Lévy flights. A summary of the discussion may be found in a recent work [A.V. Milovanov and A. Iomin, Phys. Rev. E. 107, 034203 (2023)].