Long-time stability for nonlinear Maryland models

Abstract

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

The result is covered by a submitted preprint (not on arXiv)