The Kinks, the Solitons and the Shocks in Series-Connected Discrete Josephson Transmission Lines

The localized running waves in the discrete Josephson transmission lines (JTL), constructed from Josephson junctions (JJ) and capacitors, are analytically studied. The quasicontinuum approximation reduces calculation of the running wave properties to the problem of equilibrium of an elastic rod in the potential field. Making additional approximation, the problem is reduced to the motion of the fictitious Newtonian particle in the potential well. It is shown that there exist running waves in the form of supersonic kinks and solitons and their velocities and profiles are calculated. It is shown that the nonstationary smooth waves which are small perturbations on the homogeneous nonzero background are described by Korteweg–de Vries equation and those on zero background by modified Korteweg–de Vries equation. The effect of dissipation on the running waves in JTL is also studied and it is found that in the presence of the resistors, shunting the JJ and/or in series with the ground capacitors, the only possible stationary running waves are the shock waves, whose profiles are also found. Finally, in the framework of Stocks expansion, the nonlinear dispersion and modulation stability in the discrete JTL are studied.