Subdiffusion in classical and quantum nonlinear Schrödinger equations with disorder

Abstract: The review is concerned with the nonlinear Schr\"odinger equation (NLSE) in the presence of disorder. Disorder leads to localization in the form of the localized Anderson modes (AM), while nonlinearity is responsible for the interaction between the AMs and transport. The dynamics of an initially localized wave packets are concerned in both classical and quantum cases. In both cases, there is a subdiffusive spreading, which is explained in the framework of a continuous time random walk (CTRW), and it is shown that subdiffusion is due to the transitions between those AMs, which are strongly overlapped. This overlapping being a common feature of both classical and quantum dynamics, leads to the clustering with an effective trapping of the wave packet inside each cluster. Therefore, the dynamics of the wave packet corresponds to the CTRW, where the basic mechanism of subdiffusion is an entrapping of the wave packet with delay, or waiting, times distributed by the power law $w(t)\sim 1/t^{1+\alpha}$, where $\alpha$ is the transport exponent. It is obtained that $\alpha=1/3$ for the classical NLSE and $\alpha=1/2$ for the quantum NLSE.