Phase locking in Josephson ladders and the discrete sine-Gordon equation: the effects of boundary conditions and current-induced magnetic fields.

We report on the stability of phase-locked solutions to ladder arrays of underdamped Josephson junctions under both periodic and open boundary conditions and in the presence of current-induced magnetic fields. We calculate the Floquet exponents based on the resistively and capacitively shunted junction (RCSJ) model, as well as on a simplified model of the ladder that leads to a discrete sine-Gordon (DSG) equation for the horizontal, current-biased junctions. In the case of zero induced magnetic fields, we find the DSG equation (commonly applied to parallel arrays) appreciably overestimates the exponents of the full ladder in the overdamped regime (corresponding to the limit of small junction capacitance, beta(c)), and that difference physically results from differing spectra for small-amplitude phase oscillations of the DSG and RCSJ equations. mutual inductance between plaquettes is included we find there are ranges of values for the mutual inductance for which the ladder is in fact unstable. To understand the cause of the observed instabilities, it is crucial to consider the behavior of the vertical junctions.