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Investigation reveals exponential decay and quasi-periodic solutions in Anderson localization, indicating significant implications for nonlinear dynamics.
In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model H=εΔ+π(θ+j·α)δj,j' on Zᵈ. Specifically, if ε,δ are sufficiently small, we construct a large number of time quasi-periodic and space exponentially decaying solutions (i.e., Anderson localized states) for the equation i∂ u/∂ t=Hu+δ|u|²ᵖu with a Diophantine α. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.
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Liu et al. (2025) studied this question.
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