Leapfrogging dynamics of interacting solitons in weakly coupled nonlinear transmission lines

Abstract

The dynamics of two electrical pulses forming a boundstate, propagating along two nonlinear transmission lines weakly coupled by linear capacitors shunted with linear resistances, is considered from both analytical and numerical standpoints. The study rests on an analysis of time series of the amplitudes and phases of the two interacting electrical pulses, within the framework of the variational theory based on exact one-soliton solution to the Korteweg-de Vries equation. In the regime where the two pulses propagate at nearly equal velocities, their relative amplitude/phase evolutions can result in periodic quasi-harmonic oscillations so-called leapfrogging motion. In this specific regime of motion, it is found that besides the expected damping effect on the soliton amplitudes, the resistance can also sustain their leapfrogging motion. Analytical expression of the leapfrogging frequency is derived, providing a better understanding of the competing effects of the coupling capacitor and the resistive shunt on the leapfrogging motion. Leapfrogging motions of co-propagating pulses in electrical networks can be very useful in high-intensity signal transmissions involving least energy cost for the propagating signals.