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

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.

arXiv comment38 pages (1.5 spacing), 6 figures

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Cited by in corpus (3)

• Collective vibrations of a hydrodynamic active lattice
• Statistical self-organization of a gas of interacting walking drops in a confining potential
• The invariant measure of a walking droplet in hydrodynamic pilot-wave theory