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

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