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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.
Highlights
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
In a previous study [31], we investigated the leapfrogging dynamics of soliton pairs propagating along two LC nonlinear transmission line (NLTL), weakly coupled by a linear capacitance shunted with a linear resistance
We have investigated the leapfrogging dynamics of a pair of Korteweg–de Vries (KdV) solitons in two nonlinear transmission lines, weakly coupled by a linear capacitance
Summary
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]. The coupled set of KdV equations is 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
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