Travelling waves in discrete electrical lattice with nonlinear symmetric capacitor

We study the propagation of voltage in a model of the conventional right-handed transmission line with a nonlinear symmetric capacitor. Applying the quasidiscrete approximation to the nonlinear voltage equation of the line, we derive a nonlinear Schrödinger equation and find the bright and dark solutions which are used as the initial condition for the integration of the nonlinear lattice model. The full integration of the lattice shows the propagation of the nonlinear voltage on the right-hand side and its robustness in time. The density energy of the lattice at each time has a node around which bright voltage is localized while in the case of dark voltage, it has a node where it seems to drop to zero at the soliton center. The plane phases allow proposing the characteristic of bright and dark solitons around the zero voltage.


Introduction
The use of discrete nonlinear electrical transmission lines as a theoretical or experimental tool to investigate soliton properties has been the focus of considerable research since the pioneering works by Hirota and Suzuki [1,2]. The ability of solitons to propagate with small dispersion can be used as an effective means to transmit data, modulated as short pulses over long distances justify in part its grown interest. Solitons find an application in many areas including fibre-optic communications, Bose-Einstein condensation, DNA double-strand modelling and electronic circuits [3,4]. Nonlinear transmission lines (NLTLs) which are a part of the latter mentioned area have become extremely active in recent years. This is due in part to the fact that they can be characterized using inexpensive laboratory equipment such as function generators and oscilloscopes; the availability of strongly nonlinear circuit elements. NLTLs are used for parametric amplification and pulse generation [5], for the management of localized energy [6][7][8][9][10], for nerve models with self-excitable membrane [11,12], for the studies of diatomic lattice solitons [13,14] and as a model of Microtubule [15], for the study of double-negative metamaterial [16], for the development of Shockwaves [17], for the terahertz frequency generation [18], for Short Pulse Amplification [19], just to mention some examples.
It is known that solitary wave results from the balance of dispersion and nonlinearity effects. In NLTLs the nonlinearities come generally from the voltage dependence of the resistor, inductor or capacitor. In the case where the nonlinearity comes from the capacitor, several forms have been studied. One can mention the logarithmic function used in the Toda lattice [20][21][22][23][24][25][26][27] in which Marquié et al. have observed experimentally localized mode of the electrical lattice; the exponential [6] or the results of the Taylor expansion of an experimentally double exponential [28] function. In the case where the nonlinear capacitance is the voltages reverse-biased diode, its expression has an asymmetric quadratic expression as in refs. [29][30][31][32][33][34][35].
More recently, the fabrication, measurement, and modelling of radio-frequency, tunable interdigital capacitors with ferroelectric barium strontium titanate [36,37] have opened the way for new studies in the electrical line. The voltage-dependent, symmetric capacitance of the material developed allows for the construction of nonlinear left-handed metamaterials electrical line which supports rogue waves [38] and bright breathers [39]. About the conventional transmission line which adopts normal dispersion, the study of the behaviour of solitons has not yet done with this new type of nonlinear capacitor to the best of our knowledge. Exploring the possibility of travelling solitons with the symmetric capacitor in a right-handed transmission line is the main purpose of this work.
In the next section, we present the theoretical model, the energy of the lattice, and the linear and nonlinear analysis. Numerical results are presented in Sect. 3. In Sect. 4 we summarize our results and conclude.

Model description
Following Refs. [20][21][22][23][24][25][26], let us consider a unit cell of a dispersive nonlinear transmission line shown in Fig. 1 as our model under investigation. Each unit cell, such as the nth one, contains a linear inductor L 1 in the series branch, and a linear inductor L 2 in parallel with a nonlinear capacitor C(V n ) in the shunt branch. V n (t) denotes the voltage across the capacitor whose capacitance is C(V n ), I n the current through the inductor in the series branch, I ′ n the current through C(V n ) and I ′′ n the current through the inductor in the shunt branch. Applying Kirchhoff's laws in the nth section leads to the following set of conservation laws of the current and the charge: where Q n denotes the charge in the nonlinear capacitor and allows to determine the type of nonlinearity within the lattice. The nonlinear capacitor can be in the form of Toda lattice [20][21][22][23][24][25][26][27] or its expression can be obtained from the Taylor expansion of an experimentally double exponential form [28]. In that case, the nonlinear capacitance of the diode varactor has an asymmetric quadratic expression as in the low voltages reverse-biased diode [29][30][31][32][33][34][35]. The choice of charge here is motivated by the recent development of strongly nonlinear and voltage symmetric barium strontium titanate thin film capacitors and the charge along the capacitance is Q n = C(V n ).V n . The capacitor is given by [37]: where C 0 is the zero bias capacitance and V 0 the "2:1" voltage [37]. The dependence of the nonlinear capacitor is depicted by the black curve in Fig. 2. The symmetry of the capacitor can be observed. For simplicity and by analogy with the saturable capacitor [8], will assume a Taylor expansion of the capacitor (5) (see the red curve in Fig. 2) and obtain the polynomial charge as in Refs [38][39][40] as follows: . his form of the nonlinear charge can be also found in the split-ring resonators. [41]. Using equations (1), (2), (3), (4) and (6), we can obtain the discrete voltage propagation equations of the lattice given by: where are the characteristic frequencies of the lattice. For the numerical experiment, the following parameters are used [21]

Energy of the lattice
The local energy in the different capacitive and inductive elements will be derived following refs [39,40] as: The current I n and I ′′ n will be derived numerically using equations (1) and (2), respectively.

Linear analysis: linear dispersion and group velocity
Linear oscillation in the lattice with angular frequency and wave number k is described by the following linear dispersion law: The linear dispersion curve corresponding to Eq. (10) is depicted in Fig. 3. It appears a lower cutoff mode frequency f 0 = 0 ∕2 a t k = 0 a n d a n u p p e r-f r e qu e n c y The latter is a consequence of the discretization of the lattice; this means that it does not exist in the continuum.

The Nonlinear Schrödinger equation and solutions
Due to the symmetric nature of the capacitor, only one harmonic in the semi-discrete rotating-wave approximation can be used as the solution of Eq. (7). Let us consider the solution as follows: Group velocity (11) where n = kn − t and stands for a complex unknown envelope function depending on the slow scales x = (n − V g t) and = 2 t . Inserting expression (12) in (7), we obtain the dispersion relation (10) and the group velocity (11) by keeping terms of exp(−i t) and 2 exp(−i t) respectively. Finally, we obtain a nonlinear evolution equation for the unknown function namely, the NLS equation by keeping the terms of order 3 exp(−i t) as follows: with dispersion and nonlinearity coefficients, P and Q, respectively, given by the following expressions: The soliton solutions of Eq. (13) depend on the sign of the product of the dispersion and nonlinear coefficients PQ. Then we have to analyze the behavior of PQ in the function of the frequency within the first Brillouin zone. Figure 5 shows the dependence of the dispersion coefficient (14) as a function of the frequency ( f 0 ≤ f ≤ f max ): it appears that P > 0 for f ∈ [f 0 , f s [and P < 0 for f ∈]f s , f max ] . There is a particular frequency f s for which P = 0. This frequency induces maximum group velocity (see Fig. 4) and implies the resonance of the nonlinearity coefficient Q. The dispersion which generally tends to prevent wave steepening does not exist and Eq. (13) does not allow soliton solution. The nonlinear (12) V n (t) = (x, ) exp(i n ) + * (x, ) exp(−i n ), coefficient is depicted in Fig. 6 and on can observe the negative value of Q as a function of the frequency within the first Brillouin zone. Figure 7 depicts the dependence of the product as a function of the frequency. The shaded area represents the band of the frequency ( f s ≤ f ≤ f max ) for which PQ > 0 and the other area ( f 0 ≤ f ≤ f s ) corresponds to PQ < 0 . The lattice under study has one zone of the bright soliton and another for dark soliton as in the vertical dust grain oscillations in dusty plasma crystals [56]. The continuous Schrödinger equation (13) supports the following known bright soliton solution [57] for f s ≤ f ≤ f max and the dark solitons solution for f 0 ≤ f ≤ f s . In Eqs. (16) and (17), is the soliton's amplitude, = √ | Q 2P | is the inverse width of the solitons, K is the soliton's wave number, V is the soliton's velocity and Ω = (1∕2)(K 2 − 2 ) the frequency. Following the solutions (16) and (17), the unknown voltages V n (t) in equation (7) in terms of the original coordinates can be given, respectively, by: where V in is the soliton's amplitude and Ω 0 the frequency given, respectively, by:

Numerical experiments
The numerical simulations of this section consist of the integration of the exact discrete equation (7) governing the voltage propagation along the lattice, the local (9)and total energies (9) of the line with the experimental parameters given by Eq. (8). The parameters related to the soliton's amplitude are = 0.021 and = 1 . Figure 3 shows that the frequency is in order of 10 6 Hz. That means that the integration time is in order of 10 −6 seconde ( s ). The numerical experiment will be performed with 300 s as the maximum experimental time.
Any frequency chosen in the phonon band transmission allows obtaining the wave number k using Eq. (10), the group velocity using Eq. (11), the dispersion coefficient using Eq. (14), the nonlinear coefficient using Eq. (15) and the initial voltage gives by Eqs. (18) and (19). For every chosen frequency f in the pass band ( f 0 ≤ f ≤ f max ), the wave vector k can be determined using the dispersion relation (10) as: k = 2 arcsin √ 4 2 f 2 − 2 0 4u 2 0 and the group velocity using (11). Knowing the frequency, wave vector and the group velocity, it is easy to obtain the value of the dispersion  7) for the frequency f = 0.42 MHz and Eq. (18) (for t = 0 ) as the initial condition. bThe evolution of bright voltage in cell 10. c Spatiotemporal propagation of the local energy (9). d The energy density of the bright voltage computes at t = 100 μs ▸ coefficient P (14) and the nonlinear coefficient Q (15) whose values will be used within the initial conditions.

Bright soliton propagation
It is important to point out from the experimental point of view that bright soliton has been realized with electrical lattice with asymmetric nonlinear capacitor [29] and logarithmic capacitor name as Toda lattice [21]. Here we are interested in the numerical experiment of the lattice with a symmetric capacitor. The initial condition of the bright soliton in our numerical experiment is given by the analytical solution (18).
In Fig. 8 a), we depict the spatiotemporal contour plots of the discrete voltage of the lattice governed by Eq.(7) for the frequency f=0.62MHz and Eq. (19) (for t=0) as the initial condition. As our lattice is a conventional transmission line (parallel group and phase velocities ), the wave travel on the right-hand side. Figure 8 b) shows the time evolution of bright voltage in cell 10 and one can observe that the initial pulse maintains its shape. That demonstrates the robustness of the bright soliton solution in the lattice. The modulation of the voltage within the time is observed. Its constant amplitude confirms its stability. Figure 8 c) depicts the spatiotemporal propagation of the local energy (9). Firstly, it is evident to observe the propagation of the energy of the bright on the right-hand side. This is in agreement with the positive group velocity and the increasing dispersion relation with the wave number. Secondly, there is a localization of energy in the middle of the lattice. This is confirmed by the spatial distribution of the local energy along the line at the integration time t = 20 s given in Fig. 8d. The left-handed nonlinear transmission line with the Josephson junction [58] has also the localization on the spatial distribution of the local energy observed there but with energy higher than that obtained here. Figure 9 depicts the phase plane of the bright soliton corresponding to our line. The modulation is observed around zero voltage with the maximum equal to the amplitude of the bright soliton obtains in Fig. 8b. The inset shows the behavior around zero voltage. This is similar to the behavior of the manifold around the linearly stable center (see ref [24]) for the steady stage of the voltage. This result may help from now as a characteristic of bright soliton in the non-stationary phase plane.

Dark soliton propagation
For the dark soliton propagation, the initial condition used is Eq. (19) (for t = 0 ) for frequency f s ≤ f ≤ f max . Figure 10a shows the spatiotemporal evolution of the dark voltage of Eq. (7) with frequency f = 0.42 MHz. One can see the robustness of the dark voltage within the lattice and the propagation on the right-hand side confirming the fact that our line is the conventional right-handed transmission line. The evolution of dark voltage in cell 10 is depicted in Fig. 10b and modulated dark voltage as a function of time is observed. In Fig. 10c, we depict the spatiotemporal evolution of the local energy (9). In agreement with the left-handed nonlinear transmission line with the Josephson junction, local energy drop to zero at the soliton center (see Fig. 10d). Figure 11 displays the phase plane of the dark soliton corresponding to lattice under study. The modulation of waves around the zero voltage is observed. The inset shows the behavior around the zero voltage is similar one of the manifold around an unstable saddle-node for steady state equation of the voltage. The difference with the steady state is that there is no heteroclinic orbit. Future studies will allow confirming this characteristic of the modulated dark soliton.

Conclusions
In summary, we have studied analytically and numerically discrete voltage in an electrical lattice with a nonlinear symmetric capacitor. In the linear approximation, the angular frequency increases with the wave number and induces a positive value of the group velocity and predicts the forward waves. By studying analytically the nonlinear lattice using the quasidiscrete approximation, we derived in the small amplitude approximation the nonlinear Schrödinger equation and found that the system supports bright soliton with constant width over time and stable dark soliton. The numerical experiment shows that both voltage and energy travel on the right-hand side. The plane phases allow seeing that the bright soliton has the same behavior as the manifold  [59] to solve NLS equation modelling electric lattice in the future. We expect that the results presented in this work may be useful for the enhancement of microwave devices and for phenomena of travelling robust waves.