The hidden role of coupled wave network topology on the dynamics of nonlinear lattices

TL;DR

This paper introduces a formalism for Fourier transforming network structures in nonlinear lattices, revealing new topological insights and control mechanisms for their dynamics by analyzing coupled wave networks.

Contribution

It develops a novel Fourier-based formalism for coupled wave networks, unifying various nonlinear lattice models and uncovering new dynamical regimes and topological features.

Findings

Unified nonlinear lattice models through Fourier transform

Identified new topological regimes in nonlinear media

Demonstrated control of system dynamics via network topology

Abstract

In most systems, its division into interacting constituent elements gives rise to a natural network structure. Analyzing the dynamics of these elements and the topology of these natural graphs gave rise to the fields of (nonlinear) dynamics and network science, respectively. However, just as an object in a potential well can be described as both a particle (real space representation) and a wave (reciprocal or Fourier space representation), the `natural' network structure of these interacting constituent elements is not unique. In particular, in this work we develop a formalism for Fourier Transforming these networks to create a new class of interacting constituent elements $-$ the coupled wave network $-$ and discuss the nontrivial experimental realizations of these structures. This perspective unifies many previously distinct structures, most prominently the set of local nonlinear lattice models, and reveals new forms of order in nonlinear media. Notably, by analyzing the topological characteristics of nonlinear scattering processes, we can control the system's dynamics and isolate the different dynamical regimes that arise from this reciprocal network structure, including the bounding scattering topologies.