Re-localization due to finite response times in a nonlinear Anderson chain

We study a disordered nonlinear Schrödinger equation with an additional relaxation process having a finite response time $τ$. Without the relaxation term, $τ=0$, this model has been widely studied in the past and numerical simulations showed subdiffusive spreading of initially localized excitations. However, recently Caetano et al.\ (EPJ. B \textbf{80}, 2011) found that by introducing a response time $τ> 0$, spreading is suppressed and any initially localized excitation will remain localized. Here, we explain the lack of subdiffusive spreading for $τ>0$ by numerically analyzing the energy evolution. We find that in the presence of a relaxation process the energy drifts towards the band edge, which enforces the population of fewer and fewer localized modes and hence leads to re-localization. The explanation presented here is based on previous findings by the authors et al.\ (PRE \textbf{80}, 2009) on the energy dependence of thermalized states.

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