Localized state for nonlinear disordered stark model

TL;DR

This paper proves the existence of localized, quasi-periodic states in a nonlinear disordered Stark model, demonstrating that such states persist under certain conditions and exhibit power-law decay.

Contribution

It introduces a novel analysis combining linear operator diagonalization and KAM theory to establish localization in a nonlinear disordered Stark model.

Findings

Existence of time quasi-periodic localized states for most disorder realizations.

Localized states exhibit arbitrary power-law spatial decay.

Results hold for a range of parameters elta and psilon.

Abstract

In this paper, we consider the following nonlinear disordered Stark model: $${\bf i}\partial_tu_n+\delta(u_{n+1}+u_{n-1})+nu_n+v_nu_n+\epsilon |u_n|^{2}u_n=0,\quad n\in\mathbb{Z}.$$ By employing the diagonalization of the associated linear operators and the KAM theory for nonlinear Hamiltonian systems, we establish that for parameters $\delta$ and $\varepsilon$ 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.