Characterization of leapfrogging solitary waves in coupled nonlinear transmission lines

Abstract

Leapfrogging solitary waves are characterized in two capacitively coupled transmission lines that are periodically loaded with Schottky varactors, called coupled nonlinear transmission lines (NLTLs). The coupling implies that a nonlinear solitary wave moving on one of the lines is bounded with the wave moving on the other line, which results in the periodic amplitude/phase oscillation called leapfrogging. In this study, we clarify how the leapfrogging frequency depends on the physical parameters of coupled NLTLs using a numerical model validated through measuring test lines and demonstrate the relaxation of leapfrogging. In addition, coupled Korteweg-de Vries equations are derived by applying the reductive perturbation method to the transmission equations of coupled NLTLs. Using perturbation theory based on the inverse scattering transform, a closed-form expression of leapfrogging frequency is obtained and the parameter values that simulate the properties well are examined. Engineering applications based on leapfrogging are finally discussed.

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

“…The two lines are coupled by means of linear capacitor C m at each mode. The nonlinear capacitors are varactor diodes, and except for their opposite polarities in our study, we shall use a common type of varactor diode with a Schottky barrier for both lines [30,31]. Therefore, our Schottky varactors can be represented by the following capacitance-voltage characteristics [41]:…”

“…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.…”

Modulation of Pulse Train Using Leapfrogging Pulses Developed in Unbalanced Coupled Nonlinear Transmission Lines

“…For weak couplings, the leapfrogging solitary waves are well modeled by the coupled Korteweg-de Vries (KdV) equations. Recently, we investigated two identical transmission lines with regularly spaced Schottky varactors coupled via capacitors, called coupled nonlinear transmission lines (NLTLs), and successfully observed leapfrogging phenomena for the nonlinear solitary waves developed in them [7]. Because of easiness in designing both nonlinearity and dispersion separately, the electronic system can characterize leapfrogging waves efficiently.…”

“…The phase and amplitude of leapfrogging pulses depend on the initial delay between incident pulses, such that the pulse amplitude at the output port varies with the initial delay. Accordingly, the temporal delay between two inputted pulses is converted to the pulse amplitude at the output [7]. Another potential of electrical leapfrogging pulses results from their management by the biasing voltage to the varactors.…”

“…The major output is thus switched from one line to the other by varying the bias voltage. Hence, the leapfrogging in coupled NLTLs can provide a novel switching method for the incident pulse by which the incident pulse is selectively output to one of the two ports of the coupled NLTLs [7]. The same mechanism can be used to modulate the inputted pulse train by the biasing voltage.…”