Shock wave in series connected Josephson transmission line: Theoretical foundations and effects of resistive elements

We analytically study the shock wave in the Josephson transmission line (JTL) in the presence of ohmic dissipation. When ohmic resistors shunt the Josephson junctions (JJ) or are introduced in series with the ground capacitors, the shock is broadened. When ohmic resistors are in series with the JJ, the shock remains sharp, as it was in the absence of dissipation. In all the cases considered, ohmic resistors do not influence the shock propagation velocity. We study an alternative to the shock wave—an expansion fan—in the framework of the simple wave approximation for the dissipationless JTL and formulate the generalization of the approximation for the JTL with ohmic dissipation.