Dynamics of an underdamped Josephson-junction ladder

We show analytically that the dynamical equations for an underdamped ladder of coupled small Josephson junctions can be approximately reduced to the discrete sine-Gordon equation. As numerical confirmation, we solve the coupled Josephson equations for such a ladder in a magnetic field. We obtain discrete-sine-Gordon-Like IV characteristics, including a flux how and a ''whirling'' regime at low and high currents, and voltage steps that represent a lock-in between the vortex motion and linear ''phasons,'' and which are quantitatively predicted by a simple formula. At sufficiently high anisotropy, the fluxons on the steps propagate ballistically.