† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11174145 and 11334005) and the Research Foundation for Young Scientists of Anhui University of Technology (Grant No. QZ201318).

We numerically investigate the excitation of soliton waves in the nonlinear electrical transmission line formed by many cells. When the periodic driving voltage with frequency in the pass band closing to the cutoff frequency is applied to the endpoint of the whole line, the soliton wave can be generated. The numerical results show that the soliton wave generation mainly depends on the self modulation associated with the nonlinear effect. In this study, the lower subharmonic component is also observed in the frequency spectrum. To further understand this phenomenon, we study the dependence of the subharmonic power spectrum and frequency on the forcing amplitude and frequency numerically, and find that the subharmonic frequency increases with the gradual growth of the driving amplitude.

As is well known, in the past few decades, the nonlinear phenomenon has received a great amount of attention. In particular, the soliton has aroused considerable interest ever since its introduction by Benzi *et al.* in 1981.^{[1]} The soliton can be viewed as representing a balance between the effect of nonlinearity and that of dispersion.^{[2]} In other words, this phenomenon can be found to result from the dynamic interaction between nonlinearity and dissipation. In Ref. [3], the generation of a hole soliton was discussed. Based on the nonlinear Schrödinger equation, the envelope and hole solitons have been also studied well.^{[4]} In recent years, the gap soliton has been progressively reported in a broad variety of nonlinear systems which have periodic structures.^{[5–9]} By means of nonlinear supratransmission,^{[10,11]} Geniet and Leon have investigated the local structure inside the forbidden band, and have given the concept of the cutoff soliton. In addition, the interaction between two solitary waves has been studied in Hertzian chains,^{[12]} and a family of exact solutions describing discrete solitary waves have been presented^{[13]} in a nonintegrable Fermi–Pasta–Ulam (FPU) chain. By using the electromagnetically induced transparency effect which produced high dispersion and nonlinearity, Du *et al.* have investigated the formation environment and the evolution of the dark soliton with environment parameters.^{[14]} For cases below and above the critical value, bright and dark soliton states admitted by the magnon density distribution have been studied respectively, and the two-soliton collision properties that are modulated by the current have also been discussed.^{[15]} As one of the nonlinear phenomena, the modulational instability has been investigated.^{[16,17]} It can be viewed as the interaction between the nonlinear effect and the dispersive effect. In Ref. [16], the envelope soliton can be induced by the modulational instability that leads to a self induced modulation.

The nonlinear phenomenon has been studied in diverse fields, such as nonlinear optics, fluid dynamics, plasma physics, and the electrical transmission line. Indeed, as a convenient tool, the nonlinear *LC* transmission line (LCTL), which has *N* cells consisting of inductors *L* and nonlinear capacitors *C*, plays a key role in the research of nonlinear wave propagation. Since the pioneering work by Hirota and Suzuki^{[18]} on a simulation line of the Toda lattice,^{[19]} it has received a great amount of attention in the nonlinear excitation behaviour investigations. In particular, it provides a useful way to study how the wave propagates in a nonlinear dispersive medium and to model the exotic properties of other new systems.^{[2,20,21]} For example, the transmission line can be reduced to the nonlinear Schrödinger model.^{[22]} As the Frenkel–Kontorova and the FPU models are used to investigate the thermal asymmetric energy flux,^{[23–29]} the LCTLs have been used for modulation instability and the gap soliton,^{[7,9,16,22]} and great progress has been achieved.

In view of the soliton investigation with the driving frequency above the cutoff, quite a natural question then arises whether the soliton wave would still occur when the driving frequency is chosen in the pass-band. Indeed, while the linear wave is propagating in the system, the modulational instability would be generated by the nonlinearity and dispersion, and then the self modulation (SM) would be induced. In an optical system, the SM and the soliton excitation have been studied.^{[30]} In this paper, we will choose the appropriate driving amplitude and frequency, and numerically investigate this interesting phenomenon in the LCTL. The outline of the paper is as follows. In Section 2, we will present the model first, which is, in fact, a nonlinear LCTL consisting of many identical cells. In Section 3, we numerically investigate the soliton excitation on the nonlinear electrical network. The dependence of the subharmonic power spectrum and frequency on forcing parameters will be discussed in Section 4. Finally, Section 5 is devoted to some concluding remarks.

The system under consideration is a lossless nonlinear discrete LCTL which is comprised of many elementary cells as shown in Fig. *L* in the series branch and a capacitor *C*(*V*) in the parallel branch. This model has already been used extensively to investigate the cutoff soliton^{[22]} and the asymmetric energy flux phenomenon^{[28,29]} both theoretically and experimentally. *V*_{n} represents the voltage across the *n*-th capacitor, while *I*_{n} denotes the current through the inductor *L* of the *n*-th cell with the subscript *n* designating the cell number in the network. In the following analysis, we assume that the characteristic of each cell between the capacitance *C*(*V*_{n}) and voltage *V*_{n} is described by^{[2]}

*C*

_{0}= 320 pF is the linear capacitance at the biased voltage −2 V, and

*α*= 0.21 V

^{−1}and

*β*= 0.0197 V

^{−2}are the nonlinear coefficients.

Applying the Kirchhoff’s laws and the capacitance–voltage relationship represented by Eq. (*V*(*n*) propagation:

*kn*−

*ωt*)], where

*ω*is the frequency of the wave with wave number

*k*. By inserting the general solution into Eq. (

*ω*

_{c}is the cutoff angular frequency, and

*ω*

_{c}= 2

*ω*

_{0}. Obviously, the system is a typical low pass filter. For our LCTL, the parameter of the inductance

*L*is set to be 220 μH, so that the numerical value of the upper cutoff frequency is

*f*

_{c}=

*ω*

_{c}/2

*π*= 1.1997 MHz. The linear dispersion curve of Eq. (

In the study, in order to generate the soliton wave, the *f*_{s} is the lower frequency which can be chosen by the driving one, and the magnitude of *f*_{s} is 1.0403 MHz. Namely, the driving frequency satisfies

In Refs. [16] and [22], the dispersion curve (see Fig. *P*(*ω*) and nonlinear coefficient *Q*(*ω*) have also been obtained as

In the whole area, the *P*(*ω*) is always negative, whereas the *Q*(*ω*) can be positive or negative, depending on the carrier wave frequency *ω*. In our study, the green region selected in Fig. *P*(*ω*) × *Q*(*ω*) > 0, where the instability can occur. This is the main reason for choosing the driving frequency in the range (*f*_{s}, *f*_{c}).

In fact, in the study, the LCTL has the length *N* which is long enough to avoid the free end reflections, so that the soliton wave can well keep its propagation. Of course, the wave propagation of the internal cells still obeys Eq. (

*A*

_{0}and

*f*are the driving amplitude and frequency, respectively.

In this section, in order to generate the soliton wave, we perform the numerical simulations on the LCTL [Eq. (

The values of *L*, *C*_{0}, *α*, and *β* have been given in Section 2. The frequency and amplitude of the driving voltage are *f* = 1.1990 MHz and *A*_{0} = 0.3 V, respectively. Obviously, the driving frequency value is located in the pass band (see the green area in Fig. *t* = 1/(200*f*_{c}). To avoid initial shock, the driving amplitude *A*_{0} is fed into the LCTL smoothly. The numerical simulation is carried out for a sufficiently long time (for example, *t* = 10 ms).

Figure *A*_{0} (see red line in Fig. *V*_{max} ≈ 2.3*A*_{0}. The result shows that the wave localization can generate the amplification effect.^{[22]}

Here, we also calculate the frequency spectrum of a 100-th cell wave by using the fast Fourier transform algorithm (FFT) as shown in Fig. *f*_{t} is generated (see the inset of Fig. *f*_{d} whose value equals the driving frequency and its superharmonics (denoted by 2*f*_{d} and 3*f*_{d}) as well. The subharmonic frequency *f*_{t} obtained from Fig. *f*_{d} = 1.1990 MHz. This phenomenon shows that the slow modulation of the solitary wave oscillation occurs as shown in Fig. ^{[30]} The modulation is found to vary with the driving frequency *f* and amplitude *A*_{0}. That is to say, the different driving parameters can affect the modulation degree associated with the soliton wave generation. When the *f* is close to *f*_{s} or/and the *A*_{0} is quite small, the modulation becomes weak and the soliton wave is not easy to generate. Conversely, as *f* is increased or/and the *A*_{0} is increased, the amplitude of the spectral line will increase, and the SH component *f*_{t} will also increase. Meanwhile, in the FFT spectrum, there emerges another SH component at *f*_{h} possibly, in addition to the subharmonic frequency *f*_{t}. Figure *f* = 1.1990 MHz and *A*_{0} = 0.60 V. From the plot, we can read that *f*_{t} and *f*_{h} are about 4.10 kHz and 8.201 kHz, respectively. To generate a soliton wave, however, the driving amplitude is not increased unlimitedly. For example, when the driving frequency is set to be 1.1990 MHz, and the amplitude *A*_{0} is larger than 0.90 V, the chaos will replace the soliton wave.

In this section, we turn our attention to the dependence of SH on the driving parameters *f* and *A*_{0}. By investigation, the modulation is found to vary with *f* and *A*_{0}. That is to say, the modulation will become strong with the driving frequency *f* or (and) amplitude *A*_{0} increasing. In the spectrum, we can also clearly find the SH component. As a result of the process of the *f* or/and *A*_{0} increasing, there emerges a second SH *f*_{h} coexisting with the first one *f*_{t}. Meanwhile, the SH frequency and power would change correspondingly.

Figure *A*_{0} and the SH frequencies as the driving frequency *f* is set to be 1.1990 MHz. We observe that the values of the *f*_{t} and *f*_{h} vary with the change in the driving amplitude *A*_{0}. For *A*_{0} ≤ 0.30 V, there is only one SH component *f*_{t}, and *f*_{t} will completely depress with a gradual decrease of *A*_{0}. However, when the *A*_{0} is chosen to be in a range from 0.31 to 0.70 V, the second SH component *f*_{h} would emerge, and as the *A*_{0} increases, *f*_{t} and *f*_{h} values also increase.

The variations of SH amplitude *A* with driving amplitude *A*_{0} are calculated and shown in Fig. *f*_{t} and *f*_{h}, depend on driving amplitude *A*_{0} at the fixed frequency *f* = 1.1990 MHz. The empty and solid triangles in the figure represent the numerical data of the *f*_{t} and *f*_{h}, respectively. The full lines are the fitting curves of the numerical data. It can be observed that the amplitude of the *f*_{t} line grows rapidly with *A*_{0} changing from 0.29 V to 0.60 V. However, for 0.60 V < *A*_{0} ≤ 0.70 V, the *f*_{t} amplitude increases slowly. The *f*_{t} amplitude variation means that the strength of the temporal modulation becomes more and more strong with *A*_{0} increasing. The *f*_{h} component amplitude obeys the monotonically increasing trend similar to that of the *f*_{t} component amplitude, except that the *f*_{h} spectral line disappears completely in a range *A*_{0} < 0.30 V. As a matter of fact, we find the modulation strength becomes more and more weak in a range *A*_{0} < 0.29 V, so that all the SH amplitudes cannot be measured effectively.

Figure *f*_{t} and *f*_{h} subharmonics against the driving frequency *f* at the fixed *A*_{0} = 0.70 V. It can be seen that *f*_{t} and *f*_{h} grow progressively with driving frequency varying from 1.1600 MHz to 1.1950 MHz. When the *f* value satisfies *f* < 1.1600 MHz, the spectrum plot only has the *f*_{t} component, and with the *f* value decreasing gradually, the *f*_{t} will disappear. For 1.1600 MHz ≤ *f* ≤ 1.1800 MHz, *f*_{t} frequency increases slowly when *f* increases. However, in the range 1.1800 MHz < *f* ≤ 1.1950 MHz, *f*_{t} undergoes the first decline in the rising process in which the lowest point corresponds to *f* = 1.1900 MHz.

The relation between subharmonic amplitude *A* and the driving frequency *f* is plotted in Fig. *A*_{0} = 0.70 V. From the curves, we can see that *f*_{t} and *f*_{h} amplitudes both have monotonically increasing trends. For *f* < 1.1750 MHz, the *f*_{t} and *f*_{h} amplitudes grow rapidly. In the range 1.1750 MHz≤ *f* ≤ 1.1900 MHz, the *f*_{t} and *f*_{h} spectral lines both grow slowly.

From the above, it is easy to understand the soliton wave generation in the LCTL if we consider the nonlinear and modulation effect. Obviously, from Eq. (*α*. On the other hand, the nonlinear coefficient *Q*(*ω*) of the nonlinear Schrödinger equation is closely related to parameter *α*,^{[16,22]} namely, the variation of the *α* value can change the product *P*(*ω*) × *Q*(*ω*). In the whole analysis, the driving frequency is chosen in the special area (see the green domain of Fig. *P*(*ω*) × *Q*(*ω*) > 0 holds true, so the dependence of the subharmonic on *α* is not discussed.

We know that the higher the driving amplitude, the stronger the nonlinear effect is. When the linear wave propagates in the LCTL, the result is that the modulation is induced by the nonlinearity. The numerical data illustrate that the modulation would be enhanced by increasing the driving frequency or/and amplitude. The subharmonic components are closely related to both the driving frequency and the driving amplitude. In the system, the nonlinear effect leads to the generation of subharmonics. From the spectrum, the simulation results show that the subharmonic amplitude is dependent on the driving amplitude *A*_{0}. It indicates that the amplitude of the subharmonic will increase with an increase in the nonlinear effect.

In this work, we numerically investigate the generation of soliton waves in the simple LCTL consisting of many cells, and all the cells have the same structure and the same parameters. We can clearly see that the soliton wave can be generated in the pass band based on the SM. When the driving wave propagates in the LCTL, the SM will be induced by the nonlinear effect. This mechanism of the soliton wave generation is completely different from the one which depends on the supratransmission in the forbidden band.^{[22,29]} As a matter of fact, the choosing of the driving frequency is very important for the soliton wave generation. While the driving frequency is located in the special region of the linear dispersion curve (see the green area of Fig. *f*_{c}, and the SM will also become more and more obvious. The increase in driving amplitude, at a fixed driving frequency satisfying *P*(*ω*)×*Q*(*ω*) > 0, can also lead to a similar phenomenon. From the spectrum, we can obviously see that the SH components coexist with the fundamental one *f*_{d}. By continuously increasing *A*_{0}, the soliton wave will disappear, and be replaced by the chaos wave. We numerically investigate the dependences of the subharmonic frequency and power on the driving amplitude as well, and we find that the subharmonic frequency and power would change when the driving amplitude varies. The present numerical simulation of the generation of the soliton wave is conducted in the ideal LCTL, and the experimental investigation of this interesting work is underway, and will be reported in the future.

**Reference**