# Leapfrogging of electrical solitons in coupled nonlinear transmission lines: effect of an imperfect varactor

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## Abstract

The leapfrogging dynamics of a pair of electrical solitons is investigated, by considering two capacitively coupled nonlinear transmission lines with and without intraline resistances. We discuss two distinct transmission line set-ups: in the first, we assume two RLC ladder lines with intraline varactors and a coupling linear capacitor, and in the second, we consider two capacitively coupled lossless lines with a varactor carrying impurity (imperfect diode) in one of the two interacting transmission lines. In the first context, we find that the soliton-pair leapfrogging mimics the motion of a damped harmonic oscillator, the frequency and damping coefficient of which are obtained analytically. Numerical simulations predict leapfrogging of the soliton pair when the differences in the initial values of the amplitude and phase are reasonably small, and the resistance is not too large. In the second context, leapfrogging occurs when the impurity rate is small enough and the differences in the initial values of the amplitude as well as phase are also small. As the impurity rate increases, the soliton signal in the imperfect line gets accelerated upon approaching the defective diode, causing only this specific soliton signal to move faster than its counterpart, leading to the suppression of leapfrogging.

## Keywords

Coupled nonlinear transmission lines Soliton signals Capacitive impurity Adiabatic perturbation theory Numerical simulations## 1 Introduction

The Hirota circuit [1, 2] is a simple LC ladder circuit with a linear inductance, but an active feedback capacitor embedded within the main branch of the circuit. This circuit has long served as a paradigm for the generation and propagation of nonlinear signals in electrical networks, simulating the so-called Toda lattice [3, 4] and admitting exact soliton solutions [5, 6, 7, 8, 9]. In this electrical system, the nonlinearity balancing the dispersion (related to the ladder nature of the line) is introduced by a capacitor, whose capacitance is controlled by the imposed bias voltage, thus acting like a capacitive diode (“varicap” diode or varactor). The nonlinear signal generated in this nonlinear transmission line (NLTL) is a localized electrical signal with a bell shape, propagating with features of pulse soliton (i.e., translate at constant speed keeping a permanent bell shape) due to the effect of varactors periodically loaded throughout the line. NLTLs are of interest because of their applications in several fields, e.g., under large signal conditions NLTLs can serve as impulse compressors or frequency multipliers [10]. NLTLs have also proved to be of great practical use in extremely wideband focusing and shaping of signals [11]; in the microwave domain, they are ideal sources of highly stable large-amplitude sharp pulses [12].

Several studies have been devoted to modelling, both analytically and numerically, the propagation of nonlinear signals in NLTLs in various physical contexts including resistive NLTLs, transmission lines with impurities, networks of coupled NLTLs and so on [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Two most common theoretical pictures have emerged, namely one in which the nonlinear electrical signals are soliton solutions to the Korteweg–de Vries (KdV) or coupled KdV equations [27, 28, 29] and one in which they are looked out as modulated envelope solitons described by the nonlinear Schrödinger or coupled nonlinear Schrödinger equations [10, 12, 18, 19]. Much interesting to us, recent theoretical as well as numerical works [30, 31] have established that under specific conditions, the coupling of two NLTLs can promote novel interesting configurations of soliton bound states in which soliton pairs propagate with opposite phases, but nearly equal velocities. Bound soliton states of this kind, known as leapfrogging solitons, have actually been predicted and observed experimentally in many other physical contexts [32, 33, 34, 35, 36, 37, 38] as, for instance, in hydrodynamics and plasma dynamics.

In general, the amplitudes and phases of two leapfrogging pulses depend on their initial positions and initial velocities, such that as they propagate they remain always close one to another with their amplitude and phase differences vanishing periodically with time. In the specific context of coupled NLTLs, leapfrogging propagation of soliton pairs provides means to convey pairs of large-amplitude signals at low energy cost from their interactions. Indeed when the difference in velocities of the two signals is very small, their interaction is optimized, thus favouring a bound state in which the electrical energy will be alternately transferred from the leading soliton signal to the trailing soliton signal. This leads to a periodic change in positions of the two solitons, a leading one becoming a trailing soliton and vice versa. So to say, the leapfrogging motion can be used to manage the transmission of pairs of travelling electrical pulses in electrical networks, putting into play a minimum possible power loss from the individual electrical soliton signals.

In a previous study [31], we investigated the leapfrogging dynamics of soliton pairs propagating along two LC NLTLs, weakly coupled by a linear capacitance shunted with a linear resistance. We obtained that the inclusion of a resistive element in the shunt branch of the coupling capacitance enables to control the amplitude and phase differences of the interacting pulses during their propagation. Instructively, the coupled model considered in this previous study was an extension of the study done in ref. [29] where the author addressed the problem considering only the capacitive coupling. In the present work, we are interested in the leapfrogging motion of a soliton pair in two distinct physical contexts: first, we consider the case of two RLC NLTLs coupled via a linear capacitance and two capacitively coupled LC NLTLs one of which contains a defective varactor. We first derive, using Kirchhoff’s voltage and current rules, the discrete set of nonlinear equations for the coupled NLTLs in the two physical contexts. Next, seeking for pulse signals, a multiple-scale expansion of solutions is applied in the full continuum limit which enables us to obtain a set of coupled KdV equations in the relevant scale. The coupled set of KdV equations is then treated analytically within the framework of the adiabatic perturbation theory [39], by defining appropriate variables for leapfrogging of the two KdV pulses as they propagate at nearly equal amplitudes and velocities. Their leapfrogging are explored numerically by means of a sixth-order Runge–Kutta scheme with fixed steps [40], and conditions for suppression of leapfrogging are determined.

## 2 Analysis of leapfrogging for coupled RLC NLTLS

### 2.1 Model, line equations and coupled dissipative KdV equations

*L*in parallel with a nonlinear capacitor of capacitance \(C=C(V)\). The two lines are coupled by means of linear capacitor \(C_{m}\) at each mode.

*m*are, respectively, the zero-bias capacitance, the junction potential and the grading coefficient [41, 42]. We assume that the bias voltages in lines 1 and 2 are \(-V_\mathrm{b}\) and \(V_\mathrm{b}\), respectively, reflecting the opposite polarities of varactor diodes loaded on the two lines. For convenience, we define:

*n*th section in line 1, and \(V_{n}\) and \(I_{n}\) are, respectively, the voltage and current of the

*n*th section in line 2. In the continuum limit, when the size of elementary sections in the circuits is very small compared with the length of the transmission lines, the right-hand side of Eqs. (4) and (5) can readily be approximated with partial derivatives with respect to a continuum variable \(x=nl\). This, more exactly, corresponds to the long-wavelength approximation which consists in Taylor expanding the discrete variables \(W_{n\pm 1}\) and \(V_{n\pm 1}\), i.e.,

*W*and

*V*can be expanded in series according to:

*R*:

*z*, and scaling \(W_{1}\), \(V_{1}\), \(\tau\) and

*z*as \(W_{1} =\frac{6\gamma C_\mathrm{b}}{\alpha _0C_0}\psi\), \(V_{1} = -\frac{6\gamma C_\mathrm{b}}{\alpha _0C_0}\phi\), \(\tau = \frac{2}{\eta }T\) and \(z = \gamma u\) where \(\gamma = 1/\root 3 \of {12}\), we find:

*C*. It is worth noting that the adiabatic perturbation theory is valid only when \(C_{m}/C_\mathrm{b} \ll V_\mathrm{s}/V_\mathrm{b}\) where \(V_\mathrm{s}= 12\epsilon \kappa ^2(V_\mathrm{J}+mV_{\mathrm{b}})/\root 3 \of {12}m\) is the average voltage amplitude of the incident solitons. Given that the coupling capacitance

*C*and the intraline resistance \(R_1\) should be very small consistently with the spirit of the adiabatic perturbation theory, the leapfrogging frequency can only be increased with an increase in the pulse amplitude \(\kappa\). In the next section, we shall carry out numerical simulations on the variational Eqs. (31)–(34), in order to gain a more rich insight onto parameter values for which leapfrogging of the pulse pair is more likely to be favoured.

### 2.2 Numerical simulations of leapfrogging for the coupled RLC NLTLs

In ref. [30], an analysis of soliton leapfrogging in a model of coupled NLTLs similar to Fig. 1, but without intraline resistances, has been carried out. Much recently, we have extended the study to the context of two LC-type NLTLs coupled by a linear capacitance with a linear resistance in its shunt branch. In the present study, we shall explore numerically the influence of the intraline resistance on pulse leapfrogging. In this last purpose, we applied a sixth-order Runge–Kutta scheme [40] on the set of four coupled first-order nonlinear ordinary differential Eqs. (27)–(30). To start, we considered small initial values for \(\lambda _i\) and \(\theta _i\) and, in addition, selected very close initial values for \(\lambda _1\) and \(\lambda _2,\) on the one hand, and \(\theta _1\) and \(\theta _2,\) on the other hand, which are relevant conditions for leapfrogging to occur. Later on, we shall look at the effects of increasing the initial phase and amplitude differences, on the leapfrogging motion.

*M*, but four distinct values of the resistive coefficient

*N*, listed in the figure caption.

*N*. The amplitude difference \(\varDelta \lambda\) too oscillates harmonically in time, reflecting leapfrogging of the soliton pair. Figures 3, 4 and 5 show the numerical results obtained when the differences in the initial values of the two solitons' amplitudes and phases are increased. One sees that when the differences in the initial values of these parameters increase, their variations are more and more dominated by anharmonic oscillations. Figure 5 shows a total suppression of leapfrogging when the differences between the initial amplitudes and phases become relatively large.

## 3 Analysis of leapfrogging for coupled LC NLTLs with impurity

*x*. Again from the definition \(\mathrm{d}Q_{n}=C(W_{n})\mathrm{d}W_{n}\) and \(\mathrm{d}q_{n}=C(V_{n})\mathrm{d}V_{n}\), Eqs. (40) to (43) reduce to:

*W*and

*V*in series, i.e.,

*z*, we obtain the following equations to the order \(0(\epsilon ^3)\):

*z*as \(W_{1} =\frac{6\gamma }{\alpha _0}\psi\), \(V_{1} = -\frac{6\gamma }{\alpha _0}\phi\), \(\tau = \frac{2}{\eta _0}T\) and \(z = \gamma u\) where \(\gamma\) is the same as defined in the previous section (i.e., \(\gamma = 1/\root 3 \of {12}\)), we find:

To be more explicit, the expression of \(F(\lambda _1, u_0)\) given in formula (65) suggests that the amplitude difference will oscillate with increasing amplitude, when the defective diode is closer to the input end of the transmission line and the average amplitude \(\kappa\) of the leapfrogging solitons is not too large. When \(\kappa\) is relatively large, the impurity will accelerate the soliton signal on line 1, thus increasing its speed relative to the speed of soliton on line 2.

Concerning the issue of the effects of impurities on soliton propagation in NLTLs, it is instructive stressing that the influence of a localized impurity on soliton propagation in NLTLS has been investigated in some past works. It is therefore well established that a default-type impurity will increase the amplitude of a soliton approaching the impurity [25, 26], hence causing its acceleration in virtue of the amplitude dependence of the velocity of the KdV soliton. In recent numerical simulations, Pan et al. [26] obtained that the response of a soliton signal to the presence of a localized impurity in an LC NLTL is standard: an excess structural defect will always trap a soliton signal causing its delay, whereas a structural default will accelerate the soliton signal on approaching the impurity, whether the impurity is capacitive or inductive.

Graphs in Fig. 7 show that the soliton leapfrogging is a regular harmonic oscillation with constant maximum amplitudes when there is no impurity. However, as the impurity rate \(\beta\) (an hence the impurity coefficient \(N_1\)) is increased from zero, the amplitude difference \(\varDelta \lambda\) oscillates with the maximum oscillation amplitudes increasing with time. When \(N_1\) attains a critical value, the leapfrogging is suppressed after a short time. A look at the variations of \(\lambda _1\) and \(\lambda _2\) with time on the left graph clearly suggests that \(\lambda _1\) gets amplified after a short propagation time, but not \(\lambda _2\), implying an acceleration of the soliton signal on line 1 relative to its counterpart on line 2.

## 4 Conclusion

We have investigated the leapfrogging dynamics of a pair of KdV solitons in two nonlinear transmission lines, weakly coupled by a linear capacitance. Two different physical configurations of coupled nonlinear transmission lines were considered: the first model was two RLC lines with intraline Schottky varactors, and in the second model, we considered two coupled LC lines one of which had a localized capacitive impurity. For the first model, we obtained that adding the resistive element along with the feedback capacitor on the coupled transmission lines causes a damping of the soliton amplitudes, thus acting against leapfrogging. For leapfrogging to survive the presence of the resistive component, the average amplitude of the two interacting solitons should be large enough and the resistance relatively small, consistently with the spirit of the adiabatic perturbation theory. For the second model, we established that a defect in one of the Schottky diodes on line 1 with accelerate the soliton signal on the line, causing a drive of the second soliton with the possibility of their leapfrogging as long as the impurity rate is relatively small. As we increase the impurity rate, the soliton signal in line 1 gains in amplitude and consequently in speed, and hence cannot be followed by the soliton signal in line 2. In this case, no leapfrogging can occur.

The effects of a localized impurity on soliton signals in NLTLs have been investigated in several previous works; it is there well established that a localized impurity will always accelerate a soliton approaching the impurity when it is a structural default in the defective electrical component [25, 26]. This response of KdV soliton to the presence of a localized impurity in the NLTL is actually universal; indeed, similar behaviours are predicted in many other distinct physical systems such as Josephson junction transmission lines [44, 45], Frenkel–Kontorova systems [46, 47] and double-well systems [48].

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. A. M. Dikandé wishes to acknowledge support from the Alexander von Humboldt foundation. He also thanks Pr. Holger Kantz at MPIPKS Dresden for hosting his visit in the “Nonlinear Time Series Analysis” research group.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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