  A wavefront solution for quantum (capacitively coupled) transmission lines
with charge discreteness (PRB {\bf 64}, 235309 (2001)) is proposed for the
first time. The nonlinearity of the system becomes deeply related to charge
discreteness. The wavefront velocity is found to depend on a step discontinuity
on the (pseudo) flux variable, $f$, displaying allowed and forbidden regions
(gaps), as a function of $f$. A preliminary study of the stability of the
solutions is presented. The dual transmission line hamiltonian is proposed and
finally, we find a connection with the (quantum) Toda lattice.
