Long-time Anderson Localization for the Nonlinear quasi-periodic Schr\"odinger Equation on $\mathbb Z^d$

TL;DR

This paper proves that solutions to a nonlinear quasi-periodic Schrödinger equation on ^d remain localized for a long time, using a Birkhoff normal form transform to prevent mode transfer.

Contribution

It introduces a novel application of Birkhoff normal form to establish long-time localization for nonlinear quasi-periodic Schrödinger equations on integer lattices.

Findings

Localization persists for polynomially long times

Mode transfer is impeded by the Birkhoff normal form

Results apply to arbitrary ^d data

Abstract

Using a Birkhoff normal form transform to impede mode transfer in a finite "barrier", we prove localization of arbitrary $\ell^2$ data for polynomially long time for the nonlinear quasi-periodic Schr\"odinger equation on $\mathbb Z^d$.