A discrete complex Ginzburg-Landau equation for a hydrodynamic active lattice

TL;DR

This paper derives a discrete complex Ginzburg-Landau equation from a driven oscillator model inspired by experiments with active droplet lattices, linking nonlinear dynamics to physical properties and enabling comparison with experimental results.

Contribution

It provides a systematic derivation of amplitude equations from a physical oscillator model, connecting lattice properties to nonlinear wave phenomena in active matter.

Findings

Derivation of amplitude equations from experimental-inspired oscillator models.

Prediction of nonlinear phenomena like solitons, breathers, and waves.

Framework applicable to various active and driven oscillator systems.

Abstract

A discrete and periodic complex Ginzburg-Landau equation, coupled to a discrete mean equation, is systematically derived from a driven and dissipative oscillator model, close to the onset of a supercritical Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics---namely discrete bright and dark solitons, breathers, and traveling waves---against experiments. Further, the amplitude equations allow us to rationalize the successive bifurcations leading to these distinct dynamical states. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.