Discrete nonlinear Schr\"{o}dinger equation in complex networks

TL;DR

This paper studies how the dynamics of the discrete nonlinear Schrödinger equation are affected by complex network structures, revealing localization, delocalization, and reentrant transition phenomena as network connectivity varies.

Contribution

It introduces an analysis of DNLS dynamics on complex networks with random long-range links, highlighting how connectivity influences localization and transport properties.

Findings

Self-trapping occurs at higher nonlinearity with increased connectivity.

Localization is more prominent near fully connected networks.

Reentrant localization transition observed with varying long-range bonds.

Abstract

We investigate dynamical aspects of the discrete nonlinear Schr\"{o}dinger equation (DNLS) in finite lattices. Starting from a periodic chain with nearest neighbor interactions, we insert randomly links connecting distant pairs of sites across the lattice. Using localized initial conditions we focus on the time averaged probability of occupation of the initial site as a function of the degree of complexity of the lattice and nonlinearity. We observe that selftrapping occurs at increasingly larger values of the nonlinearity parameter as the lattice connectivity increases, while close to the fully coupled network limit, localization becomes more preferred. For nonlinearity values above a certain threshold we find a reentrant localization transition, viz. localization when the number of long distant bonds is small followed by delocalization and enhanced transport at intermediate bond numbers while close to the fully connected limit localization reappears.