The Kinks, the Solitons and the Shocks in Series‐Connected Discrete Josephson Transmission Lines

Abstract

The concept that in a nonlinear wave propagation system the various parts of the wave travel with different velocities and that wave fronts (or tails) can sharpen into shock waves is deeply imbedded in the classical theory of fluid dynamics. [1] The methods developed in that field can be profitably used to study signal propagation in nonlinear transmission lines. [2][3][4][5][6][7][8][9][10][11] In the early studies of shock waves in transmission lines, the origin of nonlinearity was due to nonlinear capacitance in the circuit. [12][13][14] Interesting and potentially important examples of nonlinear transmission lines are circuits containing Josephson junctions (JJ), [15] Josephson transmission lines (JTL). [16][17][18][19] The unique nonlinear properties of JTL allow to construct soliton propagators, microwave oscillators, mixers, detectors, parametric amplifiers, and analog amplifiers. [17][18][19] Transmission lines formed by JJ connected in series were studied from 1990s, though much less than transmission lines formed by JJ connected in parallel. [20] However, the former began to attract quite a lot of attention recently, [21][22][23][24][25][26][27][28] especially in connection with possible JTL traveling wave parametric amplification. [29][30][31] The interest in studies of discrete nonlinear electrical transmission lines, in particular of lossy nonlinear transmission lines, started some time ago, [32][33][34] but it became even more pronounced recently. [35][36][37][38] These studies should be seen in the general context of waves in strongly nonlinear discrete systems. [39][40][41][42][43][44][45] In our previous publication, [46] we considered shock waves in the continuous JTL with resistors, studying the influence of those on the shock profile. Now we want to analyze wave propagation in the discrete JTL, both lossless and lossy.The rest of the article is constructed as follows. In Section 2, we formulate the approximation to the circuit equations of the discrete lossless JTL. In Section 3, we formulate the quasicontinuum approximation and show the analogy between the problem of the running waves and the problem of equilibrium of an elastic rod in the potential field. In Section 4, by simplifying the approximation, we reduce the problem of the running waves to an effective mechanical problem, describing motion of a fictitious particle in a potential well, and study the profiles of the kinks and of the solitons. In Section 5, we consider specifically weak kinks and weak solitons. In Section 6, we discuss the effect of dissipation on the running waves in the discrete JTL. In Section 7, we formulate the modified quasicontinuum approximation and, on top of it, the simple wave approximation, which opens the way to conveniently study nonstationary waves in the JTL. In Section 8, we obtain the nonlinear dispersion law, and in Section 9, we study the modulation stability of the wavetrains. In Section 10, we briefly mention possible applications of the results obtained in the article and

Modulated Harmonic Wave in Series‐Connected Discrete Josephson Transmission Line: The Discrete Calculus Approach

“…However, it is worth comparing Figure 2 with Figure 3 borrowed from our previous publication, and showing the soliton profile calculated there. Though we didn't use there the term dark soliton previously, [1] we see that the curves are very much similar: in both cases, the amplitude goes to a constant value when x(ξ) goes to infinities and decreases in between.…”

“…In general, while carrier wave is all important in the present article, there was no such wave whatsoever in our previous publication. [1] -4 -2 0 2 4 0.5 0.6 0.7 0.8 0.9…”

“…[18] These equations describe a slowly varying wavetrain by equations determining the propagation of wave number and frequency The soliton profile calculated in ref. [1]. (Λ is the period of the JTL, which in the present publication was chosen to be equal to one.)…”

“…The circuit equations are where C is the capacitance, and is the critical current of the JJ. Differentiating Equation (1a) with respect to t and substituting from Equation (1b), we obtain the closed equation for [ 1 ] where we have introduced the dimensionless time , and .…”

“…In our previous publication [ 1 ] we considered the propagation of kinks, solitons, and shocks along the Josephson transmission line (JTL). In the present short note, we want to consider a different kind of excitations––modulated harmonic waves.…”