Anderson localized states for the nonlinear Maryland model on ${\mathbb{Z}}^d$

In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model $H=\epsilon \mathrm{\Delta }+\cot \pi (\theta +j\cdot \alpha ){\delta }_{j,j^{\mathrm{\prime }}}$ on ${\mathbb{Z}}^d$. Specifically, if $\epsilon ,\delta$ 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 $i\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=Hu+\delta \mathrm{\mid }u{\mathrm{\mid }}^{2p}u$ with a Diophantine $\alpha$. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.