Closed-Form Density of States and Localization Length for a Non-Hermitian Disordered System

TL;DR

This paper derives a closed-form expression for the density of states and localization length in a non-Hermitian disordered system, providing insights into wave localization and transmission in nonlinear optical fibers.

Contribution

It extends the Thouless formula to non-Hermitian operators and calculates the density of states for specific Gaussian pulse cases, advancing understanding of non-Hermitian disordered systems.

Findings

Derived closed-form density of states for non-Hermitian systems

Analyzed localization length in different Gaussian pulse scenarios

Discussed implications for optical fiber information transmission

Abstract

We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schroedinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of the information transmission through non-linear optical fibers.