Nonlinear transmission line: shock waves and the simple wave approximation
Eugene Kogan
TL;DR
The paper investigates nonlinear transmission lines built from nonlinear capacitors and inductors, examining shock formation under strong dissipation and introducing a simple-wave framework for lossless lines. In the strong-dissipation regime, traveling-wave reductions yield parameter-dependent ODEs with analytic shock profiles, including explicit forms such as $Q(\tau) = Q_2 + \frac{Q_1 - Q_2}{e^{\psi \tau} + 1}$ under certain constraints, and a reduced first-order relation linking $\Phi$ and $Q$. In the weak-dissipation regime, time-averaging produces stationary dispersive shocks described by Weierstrass elliptic functions, with slow evolution of elliptic invariants driven by damping. For lossless NLTL, the simple wave approximation decouples left/right traveling waves, yielding tractable equations for fully nonlinear circuits and connecting to KdV/mKdV dynamics in appropriate limits, thereby enabling analysis of kinks and solitons in nonlinear transmission lines. Collectively, the work provides analytic shock profiles, a unifying simple-wave framework, and links to classical nonlinear wave equations, with implications for pulse shaping and Josephson transmission lines.
Abstract
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. Thus, the dissipation being taken into account leads to the existence of \mbox{shocks -- the} travelling waves with 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. In the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper we consider the nonlinear lossless transmission line. We formulate a simple wave approximation for such transmission line, which decouples left/right-going waves. The approximation can also be used for the lossy transmission line, which is considered in the first part of the paper, to describe the formation of the shock wave (but, of course, not the shock wave itself).