Localized state for nonlinear disordered stark model

Abstract

In this paper, we consider the following nonlinear disordered Stark model: $$\mathbf{i}{\partial }_t{u}_n+\hat{I}´(u_{n+1}+u_{n-1})+n{u}_n+v_n{u}_n+\hat{I}\mu |u_n{|}^2{u}_n=0,n\in \mathbb{Z}.$$ By employing the diagonalization of the associated linear operators and the KAM theory for nonlinear Hamiltonian systems, we establish that for parameters $\hat{I}´$ and $\epsilon$ in a reasonable range, and for most realization of random variables $v=\{v_n{\}}_{n\in \mathbb{Z}}$, there exist time quasi-periodic and spatially localized states that exhibit arbitrary power-law spatial decay.