Nonlinear transmission line: Shock waves and the simple wave approximation

The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper, we additionally include linear ohmic resistors. The dissipation thus being taken into account leads to the existence of shocks-the traveling waves with the different asymptotically constant values of the voltage (the capacitor charge) and the current before and after the front of the wave. For the particular values of ohmic resistances (corresponding to strong dissipation), we obtain the analytic solution for the profile of a shock wave. Both the charge on a capacitor and current through the inductor are obtained as the functions of the time and space coordinate. For the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper, we consider a nonlinear lossless transmission line. We formulate the simple wave approximation for such a transmission line, which decouples left- and right-going waves. The approximation can also be used for the lossy transmission line considered in the first part of the paper to describe the formation of the shock wave (but, of course, not the shock wave itself).