Spreading in Disordered Lattices with Different Nonlinearities

TL;DR

This paper investigates how localized states spread in a nonlinear disordered lattice, using a nonlinear Schrödinger equation, and confirms the self-similar, subdiffusive nature of this spreading through entropy analysis and exponent dependence.

Contribution

It introduces a nonlinear diffusion model for spreading, characterizes the self-similar evolution with entropy measures, and links spreading exponents to nonlinearity, aligning with theoretical predictions.

Findings

Spreading exhibits subdiffusive behavior.

Structural entropy remains constant during evolution.

Spreading exponents depend on the nonlinearity index.

Abstract

We study the spreading of initially localized states in a nonlinear disordered lattice described by the nonlinear Schr\"odinger equation with random on-site potentials - a nonlinear generalization of the Anderson model of localization. We use a nonlinear diffusion equation to describe the subdiffusive spreading. To confirm the self-similar nature of the evolution we characterize the peak structure of the spreading states with help of R\'enyi entropies and in particular with the structural entropy. The latter is shown to remain constant over a wide range of time. Furthermore, we report on the dependence of the spreading exponents on the nonlinearity index in the generalized nonlinear Schr\"odinger disordered lattice, and show that these quantities are in accordance with previous theoretical estimates, based on assumptions of weak and very weak chaoticity of the dynamics.