Discrete and periodic complex Ginzburg-Landau equation for a hydrodynamic active lattice

SJ Thomson, M Durey, RR Rosales - Physical Review E, 2021 - APS
SJ Thomson, M Durey, RR Rosales
Physical Review E, 2021APS
A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is
systematically derived from a driven and dissipative lattice oscillator model, close to the
onset of a supercritical Andronov-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 …
A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is systematically derived from a driven and dissipative lattice oscillator model, close to the onset of a supercritical Andronov-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, the onset and formation of discrete dark solitons, breathers, and traveling waves—against experiments. 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.
American Physical Society
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