Characterization of partially dissipated solitons in a traveling-wave field-effect transistor

Highlights

• Solitons developed in a traveling-wave field-effect transistor are characterized.

• We investigate the KdV solitons with partial dissipation.

• Numerical calculations/experiments validate the predictions of the perturbation theory.

• Large solitons can travel without significant amplitude decay over a long distance.

Abstract

We investigate the Korteweg–de Vries equation with nonlinear dissipation, which becomes finite only for small field intensities. Using the perturbation theory based on the inverse scattering transform, we evaluate the amplitude and the phase of both one- and two-soliton solutions to show that large solitons can travel without significant amplitude decay over a long distance. We then develop a traveling-wave field-effect transistor (TWFET) that supports such partially dissipated solitons. Using both the numerical and experimental characterization of a TWFET, we validate the properties of partially dissipated solitons.

Introduction

Several interesting and functional behaviors have been discovered for the solitons described by the Korteweg–de Vries (KdV) equation when they interact with either themselves or with external objects. The width of a soliton decreases when it is amplified by coupling with an energy supplier. When interacting with a moving element, a soliton is known to be pinned by it [1]. Moreover, in a coupled KdV system, each KdV soliton is coupled with its counterpart and establishes leapfrogging propagation [2], [3], [4]. In this paper, we consider the KdV solitons that are subject to the nonlinear dissipation, which only influences them when their amplitude is smaller than a threshold value. Fig. 1(a) shows the situations we investigate. The hatched region represents the dissipation, which partially affects a soliton when its field magnitude is less than threshold $A_{\mathit{th}}$. Two solitons having different amplitudes $A_{1\text{,}2}$ are shown. It has been established that the small soliton decays significantly, while the large soliton almost preserves its shape as shown in Fig. 1(b). In this paper, we call the solitons influenced by this type of partial dissipation partially dissipated solitons.

The perturbing term causes the KdV equation to lose its integrability, so that the solution can only be characterized through suitable modifications of the soliton solutions of the unperturbed equation. With such an approach, we can sometimes estimate the perturbation effects quantitatively in the analytically closed forms. There are two main perturbation theories of solitons: the inverse-scattering-transform (IST)-based theory [5], [6], [7] and the direct perturbation theory [8], [9], [10]. Both approaches have their advantages and disadvantages. The IST-based method requires the explicit forms of the fundamental solutions of the scattering problem of the corresponding Schrödinger equation, which are sometimes difficult to obtain. However, it gives both the adiabatic and radiative dynamics. On the other hand, the direct method does not require any information other than the form of the solution. However, it cannot predict the radiative effects. In general, these two methods lead to different dynamical equations even with the first-order approximation. Moreover, there seems to be no definite way to determine which method suits the problem under investigation. Comparing the result of each perturbation theory with the corresponding physically realizable perturbed system can yield valuable information.

For the practical development of solitons in electronics, a lumped transmission line containing a series inductor and a shunt Schottky varactor in each section is utilized [11]. The line is called a nonlinear transmission line (NLTL). At present, the operation bandwidth of carefully designed Schottky varactors exceeds 100 GHz; therefore, they are employed in ultrafast electronic circuits including a subpicosecond electrical shock generator [12] and a pulse train generator implemented with an amplifier [13]. Considering these trends, the generation of short electrical pulses has universal value in high-speed electronics.

When a pulse is input into an NLTL such that the varactor nonlinearity compensates for the dispersion, the line generates multiple solitonic pulses whose widths are smaller than the input width. The NLTL operates as a good short-pulse generator by extracting the largest pulse. To obtain a shorter pulse, an NLTL must split the input into multiple pulses. Because the separation between split pulses increases only in proportion to the propagation length, the number of split pulses is limited by the line length. On the other hand, the line resistance attenuates the pulse amplitude, making the pulses too small for an NLTL to exhibit nonlinearity. This line length limitation compromises the potential of NLTLs significantly.

Because of partial dissipation, only the largest, shortest soliton among the multiple uneven ones survives after proper propagation; therefore, the partially dissipated solitons also have practical importance. To exploit the potential of NLTLs, we considered a traveling-wave field-effect transistor (TWFET). This is a special type of FET with electrodes employed as transmission lines in addition to electrical contacts [14]. It has been studied for use in broadband amplifiers or oscillators [15]. The structure is equivalently represented by two mutually coupled transmission lines with regularly spaced FETs. One of the lines is periodically connected with the gate, and the other with the drain. By introducing the Schottky diodes, the electrode line of a TWFET simulates an NLTL. Owing to the coupling between the gate and the drain lines, two different propagation modes develop in a TWFET. It is found that each mode can support solitonic waves. Moreover, a TWFET can be designed in such a way that the FET gain acts on the waves supported by one of the two modes [16]. When pulses having dissimilar parities are applied to the gate and drain lines and the gate pulse is applied in such a way that the pulse peak is below the FET threshold, partial dissipation is imparted to the solitons traveling in a TWFET.

In Section 2, we discuss the properties of partially dissipated solitons based on the IST-based perturbation theory. In Section 3, the structure and fundamental properties of a TWFET are discussed, followed by the derivation of the perturbed KdV equation that governs the partially dissipated solitons in a TWFET. In Section 4, we examine the validity of the perturbation theory by evaluating a TWFET numerically. The experimental characterization of partially dissipated solitons is discussed in Section 5.