Nonlinear Anderson Localized States at Arbitrary Disorder

Given an Anderson model \(H = -\Delta + V \) $H=-\mathrm{\Delta }+V$ in arbitrary dimensions, and assuming the model satisfies localization, we construct quasi-periodic in time (and localized in space) solutions for the nonlinear random Schrödinger equation \(i\frac{\partial u}{\partial t}=-\Delta u+Vu+\delta |u|^{2p}u\) $i\frac{\partial u}{\partial t}=-\mathrm{\Delta }u+Vu+\delta {|u|}^{2p}u$ for small \(\delta \) $\delta$ . Our approach combines probabilistic estimates from the Anderson model with the Craig–Wayne–Bourgain method for studying quasi-periodic solutions of nonlinear PDEs.