Staircase dynamics of Josephson-junction arrays

We present ansatz dynamical equations for current-driven two-dimensional square Josephson-junction arrays with magnetic flux per unit plaquette f such that the ground states form ``staircase'' configurations. For f=p/q, we need only (q+1) variables to specify a dynamical staircase state; a large reduction is thus achieved in computing time. I-V curves for f=1/3 and 2/5 obtained from these equations for dc plus ac driving are identical to those from simulations of lq×{}lq arrays. We see fractional Shapiro steps for dc plus ac current driving as well as much weaker subharmonic Shapiro steps in some circumstances. In the limit of low-frequency and low average voltage, one can further reduce the staircase equations to an approximate single-junction equation with an almost-sinusoidal supercurrent function, which is consistent with the appearance of fractional steps and weak subharmonic steps. We also find suppression of fractional harmonic stepwidths relative to integer harmonic stepwidths in the high-frequency limit.