Long-time stability for nonlinear Maryland models

For the $d-$dimensional nonlinear Maryland model \begin{equation}\label{eq-abs} \ri\partial_t q_n=\tan\pi(n\cdot\varpi+x)q_n+\epsilon(\Delta q)_n+|q_n|^2q_n,\quad n\in{\Z^d}, \end{equation} with $d\in\N^*$, $\epsilon\in \R$ and $\varpi\in\R^d$ satisfying a suitable Diophantine condition, we establish polynomial long-time stability of polynomially weighted $\ell^2$-norm $$\|q(t)\|_s:=\left(\sum_{n\in{\Z^d}}|q_n|^2 (1+|n|^2)^{s}\right)^{\frac{1}{2}},\quad s>0.$$ More precisely, given any $M_*\in\N^*$, for phase parameters $x$ belonging to an almost full-measure subset of $\R/\Z$, if $|\epsilon|$ is sufficiently small, then solutions $q(t)$ of Eq. (\ref{eq-abs}) with high-order weighted $\ell^2$-norm $\|q(0)\|_s$ of sufficiently small size $\varepsilon$ satisfy $$\|q(t)\|_s=\CO(\varepsilon),\quad \forall \ |t|\leq \epsilon^{-1}\varepsilon^{-M_*}.$$ The proof relies on a Birkhoff normal form procedure.