Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The shocks in Josephson transmission line revisited

Eugene Kogan

TL;DR

This paper investigates shocks in lossy Josephson transmission lines by deriving a continuum description and analyzing how small-amplitude sound waves scatter off shocks, yielding simple reflection and transmission coefficients. It unifies shocks and kinks within a traveling-wave framework and shows that weak kinks persist in the lossy regime, potentially forming quasi-soliton structures inside shocks; the weak-wave regime is tractable via a damped Helmholtz–Duffing reduction, with exact tanh- or front-like solutions under specific damping conditions. The work clarifies the relationship between nonlinear wave features in JTLs, providing concrete criteria for when shocks or kinks occur, and demonstrates that soliton-like profiles arise for weak losses, offering insights for nonlinear wave control in superconducting circuits. Overall, the results advance understanding of energy transport and waveform propagation in Josephson networks and lay groundwork for further analytical and experimental exploration of nonlinear Josephson-based devices.

Abstract

We continue our previous studies of the shocks in the lossy Josephson transmission line (JTL). The paper consists of two parts. In the first part we analyse the scattering of the "sound' (small amplitude small wave vector harmonic wave) on the shock wave and calculate the reflection and the transmission coefficients. In the second part we show that the kinks, which we previously studied only in the lossless JTL, exist also in the lossy JTL and study the similarities and the dissimilarities between the shocks and the kinks there. We find that the nonlinear equation describing the weak kinks and the weak shocks can be integrated (in particular cases) in terms of elementary functions. We also show that the profile of the shock in the lossy JTL demonstrates soliton-like features if the losses are weak.

The shocks in Josephson transmission line revisited

TL;DR

This paper investigates shocks in lossy Josephson transmission lines by deriving a continuum description and analyzing how small-amplitude sound waves scatter off shocks, yielding simple reflection and transmission coefficients. It unifies shocks and kinks within a traveling-wave framework and shows that weak kinks persist in the lossy regime, potentially forming quasi-soliton structures inside shocks; the weak-wave regime is tractable via a damped Helmholtz–Duffing reduction, with exact tanh- or front-like solutions under specific damping conditions. The work clarifies the relationship between nonlinear wave features in JTLs, providing concrete criteria for when shocks or kinks occur, and demonstrates that soliton-like profiles arise for weak losses, offering insights for nonlinear wave control in superconducting circuits. Overall, the results advance understanding of energy transport and waveform propagation in Josephson networks and lay groundwork for further analytical and experimental exploration of nonlinear Josephson-based devices.

Abstract

We continue our previous studies of the shocks in the lossy Josephson transmission line (JTL). The paper consists of two parts. In the first part we analyse the scattering of the "sound' (small amplitude small wave vector harmonic wave) on the shock wave and calculate the reflection and the transmission coefficients. In the second part we show that the kinks, which we previously studied only in the lossless JTL, exist also in the lossy JTL and study the similarities and the dissimilarities between the shocks and the kinks there. We find that the nonlinear equation describing the weak kinks and the weak shocks can be integrated (in particular cases) in terms of elementary functions. We also show that the profile of the shock in the lossy JTL demonstrates soliton-like features if the losses are weak.
Paper Structure (20 sections, 93 equations, 8 figures)

This paper contains 20 sections, 93 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The circuit equations: continuum approximation
  3. The sound scattering by the shock wave
  4. The sound waves and the shock waves
  5. The reflection and the transmission coefficients
  6. The shocks and the kinks
  7. The travelling waves
  8. The kinks vs. the shocks
  9. The shocks speed vs. the kinks speed
  10. The quasi-solitons within the shocks
  11. Weak kinks and elementary weak shocks
  12. Conclusions
  13. The JTL composed of superconducting grains
  14. The continuum and the discrete JTL
  15. Numerical integration of Eq. (\ref{['system26']})
  16. ...and 5 more sections

Figures (8)

  • Figure 1: Discrete Josephson transmission line.
  • Figure 2: Reflection of a sound wave from a shock wave. The horizontal axis is the coordinate $Z$, the vertical axis - instantaneous value of the Josephson phase.
  • Figure 3: Transmission of a sound wave through a shock wave. The horizontal axis is the coordinate $Z$, the vertical axis - instantaneous value of the Josephson phase.
  • Figure 4: The potential $\Pi(\varphi)$, as given by (\ref{['w']}), for $\varphi_1=.5$, $\varphi_2=.2$.
  • Figure 5: The potential $\Pi(\varphi)$, as given by (\ref{['w']}), for $\varphi_1=.5$, $\varphi_2=-.2$ (left) and for $\varphi_1=.5$, $\varphi_2=-.4$ (right). In the former case $\varphi_2$ corresponds to the minimum of the potential, in the latter - to the maximum.
  • ...and 3 more figures