Long-time stability for nonlinear Maryland models

TL;DR

This paper proves polynomial long-time stability for solutions of a nonlinear Maryland model with small perturbations, using Birkhoff normal form techniques under Diophantine conditions.

Contribution

It establishes polynomial long-time stability for the nonlinear Maryland model with small perturbations and Diophantine phase parameters, extending understanding of its dynamical behavior.

Findings

Solutions remain bounded over long times for small initial data and perturbations.

The stability time scale is polynomial in the inverse of the perturbation size.

The proof employs a Birkhoff normal form procedure.

Abstract

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.