Exact analytical solutions for the kinks, the solitons and the shocks in discrete nonlinear transmission line with nonlinear capacitance

TL;DR

This work derives exact traveling-wave solutions for discrete nonlinear transmission lines with nonlinear capacitors, covering both lossless kinks/solitons and lossy dissipative kinks/shocks. By applying a quasi-continuum reduction, the authors obtain a tractable ODE framework and explicit analytic profiles, including $V(Z)$ forms and speed relations such as $\frac{1}{U^2}=L\frac{Q_2-Q_1}{V_2-V_1}$, and show how dissipation modifies the landscape while preserving an analytically solvable structure. For cubic-type charge–voltage relations, kink and soliton solutions acquire closed forms with $E^2(V)$ factorization and hyperbolic profiles, while the dissipative case yields a solvable Abel-type condition and a phase distinction between kinks and shocks. Overall, the paper provides a coherent analytical treatment that connects discrete nonlinear transmission-line theory to classical nonlinear wave frameworks, with explicit profiles and velocity criteria applicable to Josephson/TL contexts and Landauer-type continuum limits.

Abstract

We studied discrete transmission lines constructed from ideal linear inductors and nonlinear capacitors (and possibly resistors). The localised travelling waves in the lossless transmission lines are the kinks and the solitons, which speeds and profiles were calculated. The localised travelling waves in the lossy transmission lines are the dissipative kinks and the shocks, which speeds and profiles were also calculated.

Figures (2)

• Figure 1: The lossless discrete transmission line.
• Figure 2: The lossy discrete transmission line.