Diffusion bound for the nonlinear Anderson model
Dynamical Systems
2023-03-07 v2 Mathematical Physics
math.MP
Abstract
In this paper, we prove the power-law in time upper bound for the diffusion of a 1D discrete nonlinear Anderson model. We remove completely the decaying condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud. 163: 21--42, 2007.). This gives a resolution to the problem of Bourgain (Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear disordered systems. The proof uses a novel ``norm'' based on tame property of the Hamiltonian.
Cite
@article{arxiv.2008.10171,
title = {Diffusion bound for the nonlinear Anderson model},
author = {Hongzi Cong and Yunfeng Shi},
journal= {arXiv preprint arXiv:2008.10171},
year = {2023}
}
Comments
While our Birkhoff normal form theory still works (from page 5 to page 17), we find the derivatives estimates on page 19 can not be obtained without any prior estimates of type H^a for small a>0