Universal Correlations and Dynamic Disorder in a Nonlinear Periodic 1D System

TL;DR

This paper studies how nonlinearity induces correlations and dynamic disorder in a 1D periodic system, revealing universal behaviors and phase relationships that emerge from initial random phases.

Contribution

It demonstrates that strong nonlinearity leads to universal correlation patterns and dynamic disorder in a 1D system, expanding understanding of nonlinear wave evolution.

Findings

Intensity histograms become Gaussian with strong nonlinearity

Phase correlations form between neighboring sites depending on nonlinearity sign

The correlation shape is universal, independent of system parameters

Abstract

When a periodic 1D system described by a tight-binding model is uniformly initialized with equal amplitudes at all sites, yet with completely random phases, it evolves into a thermal distribution with no spatial correlations. However, when the system is nonlinear, correlations are spontaneously formed. We find that for strong nonlinearities, the intensity histograms approach a narrow Gaussian distributed around their mean and phase correlations are formed between neighboring sites. Sites tend to be out-of-phase for a positive nonlinearity and in-phase for a negative one. The field correlations take a universal shape independent of parameters. This nonlinear evolution produces an effectively dynamically disordered potential which exhibits interesting diffusive behavior.