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Josephson transmission line revisited

Eugene Kogan

TL;DR

This work revisits the Josephson transmission line (JTL) by analyzing discrete, continuum, and quasi-continuum models of series-connected Josephson junctions with capacitors and resistors. It develops the simple-wave approximation to decouple right- and left-going waves, enabling tractable analyses of shock formation and traveling waves, including kinks and solitons, in both lossless and lossy settings. The authors derive explicit expressions for shock and soliton velocities, connect discrete-phase dynamics to continuum soliton theory (KdV/mKdV regimes for weak waves), and show how discrete and lattice effects vanish in the appropriate continuum limit. The results provide a cohesive framework for understanding compact wave propagation, energy transport, and shock formation in nonlinear JTLs with realistic dissipation and dispersion, with potential applications in microwave signal processing and nonlinear circuit design.

Abstract

We consider the series-connected Josephson transmission line (JTL), constructed from Josephson junctions, capacitors and (possibly) resistors. We calculate the velocity of shocks in the discrete lossy JTL. We study thoroughly the continuum and the quasi-continuum approximations to the discrete JTL, both lossless and lossy. In the framework of these approximations we show that the compact travelling waves in the lossless JTL are the kinks and the solitons, and calculate their velocities. On top of each of the above mentioned approximations, we propose the simple wave approximation, which decouples the JTL equations into two separate equations for the right- and left-going waves. The approximation, in particular, allows to easily consider the formation of shocks in the lossy JTL.

Josephson transmission line revisited

TL;DR

This work revisits the Josephson transmission line (JTL) by analyzing discrete, continuum, and quasi-continuum models of series-connected Josephson junctions with capacitors and resistors. It develops the simple-wave approximation to decouple right- and left-going waves, enabling tractable analyses of shock formation and traveling waves, including kinks and solitons, in both lossless and lossy settings. The authors derive explicit expressions for shock and soliton velocities, connect discrete-phase dynamics to continuum soliton theory (KdV/mKdV regimes for weak waves), and show how discrete and lattice effects vanish in the appropriate continuum limit. The results provide a cohesive framework for understanding compact wave propagation, energy transport, and shock formation in nonlinear JTLs with realistic dissipation and dispersion, with potential applications in microwave signal processing and nonlinear circuit design.

Abstract

We consider the series-connected Josephson transmission line (JTL), constructed from Josephson junctions, capacitors and (possibly) resistors. We calculate the velocity of shocks in the discrete lossy JTL. We study thoroughly the continuum and the quasi-continuum approximations to the discrete JTL, both lossless and lossy. In the framework of these approximations we show that the compact travelling waves in the lossless JTL are the kinks and the solitons, and calculate their velocities. On top of each of the above mentioned approximations, we propose the simple wave approximation, which decouples the JTL equations into two separate equations for the right- and left-going waves. The approximation, in particular, allows to easily consider the formation of shocks in the lossy JTL.
Paper Structure (25 sections, 72 equations, 5 figures)

This paper contains 25 sections, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The discrete JTL
  3. The lossless line
  4. The lossy line
  5. The continuum approximation
  6. The lossless line
  7. The simple wave approximation
  8. The lossy line
  9. The simple wave approximation
  10. The travelling waves
  11. The shocks
  12. The quasi-discrete approximation
  13. The travelling waves: the kinks and the solitons
  14. The quasi-continuum approximation
  15. The kinks and the solitons
  16. ...and 10 more sections

Figures (5)

  • Figure 1: Lossless discrete Josephson transmission line
  • Figure 2: Discrete Josephson transmission line with the capacitor and the resistor shunting the JJ and another resistor in series with the ground capacitor
  • Figure 3: The shock velocity $\overline{U}_{\text{sh}}(\varphi_1,\varphi_2)$ for $\varphi_2=0$ as function of $\varphi_1$, as given by Eq. (\ref{['velocity3']}) (solid line), Eq. (\ref{['velocb']}) (red dashed line) and Eq. (\ref{['veloc']}) (blue dot-dashed line). The green dotted line corresponds to Eq. (\ref{['hru2']}) (see Section \ref{['patash']}).
  • Figure 4: Lossless discrete Josephson transmission line: different choice of the dynamical variables.
  • Figure 5: Josephson curve given by Eq. (\ref{['kompa']}) for $\varphi_1=.5$. The solid curve represents the shocks moving to the right, the dashed -- to the left.