A reduced model for one-dimensional arrays of overdamped Josephson junctions

By assuming small inductance values and by means of fluxoid quantization, a perturbative expansion reduces the dynamical equations for the gauge-invariant superconducting phase differences in one-dimensional array containing N + 1 identical overdamped Josephson junctions to a single non linear differential equation. The resulting time-evolution equation is seen to be similar to the single-junction dynamical equation with an appropriately defined current-phase relation. This equation is coupled to N flux equations, which are solved, by means of a perturbation analysis, to first order in the parameter β. The critical current, the I- V characteristics and the flux-voltage curves of the array are determined by means of the reduced model found.