We analytically study the localized running waves in the discrete Josephson
transmission lines (JTL), constructed from Josephson junctions (JJ) and
capacitors. The quasi-continuum approximation reduces calculation of the
running wave properties to the problem of equilibrium of an elastic rod in the
potential field. Making additional approximation, we reduce the problem to the
motion of the fictitious Newtonian particle in the potential well. We show that
there exist running waves in the form of supersonic kinks and solitons and
calculate their velocities and profiles. We show that the nonstationary smooth
waves which are small perturbations on the homogeneous non-zero background are
described by Korteweg-de Vries equation, and those on zero background — by
modified Korteweg-de Vries equation. We also study the effect of dissipation on
the running waves in JTL and find 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 we study the nonlinear dispersion
and modulation stability in the discrete JTL.
