Nonlinear Anderson Localized States at Arbitrary Disorder

Given an Anderson model H = - Δ + 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 ∂ u ∂ t = - Δ u + V u + δ | u | 2 p u for small δ . Our approach combines probabilistic estimates from the Anderson model with the Craig–Wayne–Bourgain method for studying quasi-periodic solutions of nonlinear PDEs.