Nonlinear Anderson Localized States at Arbitrary Disorder

Given an Anderson model \(H = -\Delta + V \) 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 \(i\frac{\partial u}{\partial t}=-\Delta u+Vu+\delta |u|^{2p}u\) for small \(\delta \). Our approach combines probabilistic estimates from the Anderson model with the Craig–Wayne–Bourgain method for studying quasi-periodic solutions of nonlinear PDEs.