Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation

Abstract

We analyze the damped, driven, discrete sine-Gordon equation with periodic boundary conditions and constant forcing. Analytical and numerical results are presented about the existence, stability, and bifurcations of traveling waves in this system. These results are compared with experimental measurements of the current-voltage ( I–V) characteristics of a ring of N = 8 underdamped Josephson junctions. We find two types of traveling waves: low-velocity kinks and high-velocity whirling modes. The kinks excite small-amplitude linear waves intheir wake. At certain drive strengths, the linear waves phase-lock to the kink, generating resonant steps in the I–V curve. Steps also occur in the high-velocity region, due to parametric instabilities of the whirling mode. We analyze the onset of these instabilities, then numerically study the secondary bifurcations and complex spatiotemporal phenomena that occur past the onset. In all cases, the measured voltage locations of the resonant steps are in good agreement with the predictions.