Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line

We study analytically the dynamics of modulated waves in a dissipative modified Noguchi nonlinear electrical network. In the continuum limit, we use the reductive perturbation method in the semidiscrete limit to establish that the propagation of modulated waves in the network is governed by a dissipative nonlinear Schrödinger (NLS) equation. Motivated with a solitary wave type of solution to the NLS equation, we use both the direct method and the Weierstrass's elliptic function method to present classes of bright, kink, and dark solitary wavelike solutions to the dissipative NLS equation of the network. Through the exact solitary wavelike solutions to the dissipative NLS equation, we investigate the effects of the dissipative elements of the network on wave propagation. We show that the wave amplitude decreases and its width increases when the dissipative element of the network increases. It has been also found that the dissipative element of the network can be used to manipulate the motion of solitary waves through the network. This work presents a good analytical approach of investigating the propagation of solitary waves through discrete electrical transmission lines and is very important for studying modulational instability.

Figure 1

Schematic representation of the elementary cell of the lossy 1D modified discrete Noguchi electrical transmission line. The network possesses $N$ identical unit cells.

Figure 2

Linear dispersive curve of the network showing evolution of the frequency $f=\omega /2\pi$ as function of the wave number $k$ for the network parameters $L_1=200 \mu \mathrm{H}, L_2=470 \mu \mathrm{H}, V_b=2 \mathrm{V}, C_0=370 \mathrm{pF}, \alpha =0.21 {\mathrm{V}}^{-1}, \beta =0.0197 {\mathrm{V}}^{-2},$ and $C_S=50 \mathrm{pF}$. The lower cutoff frequency is $f_0=381.65 \mathrm{kHz}$ (dashed line), and the upper cutoff frequency is $f_{\max } =991.63 \mathrm{kHz}$ (dash-dotted line).

Figure 3

Signal voltage obtained with the exact bright soliton solution (27) of the dissipative NLS equation (6) for the network parameters shown in the text and for frequency $f=406.881 \mathrm{kHz}$. (a) Propagation of bright soliton signal at different cells of the network. (b) Propagation of bright soliton signal at cell $n=15$ of the network for different dissipative parameter $\sigma$. (c) Spatial propagation of bright soliton through the network at different times for $\sigma =0.00461$.

Figure 4

Propagation of bright solitary waves in the network at frequency $f=849.872 \mathrm{kHz}$ obtained with the exact bright solitary wavelike solution (28) of the dissipative NLS equation (6) for the network parameters shown in the text and the solution parameters $\varepsilon ={10}^{-2}, K=0.1, K_0=40$ and (a) $\frac{d\phi \left(\tau \right)}{d\tau }=-{10}^9$; (b) $\frac{d\phi \left(\tau \right)}{d\tau }=-{10}^9{cos}^2\left[250000{\varepsilon }^{-2}\tau \right];$ (c) $\frac{d\phi }{d\tau }=-2\times {10}^{-4}exp\left[{10}^6{\varepsilon }^{-2}\tau \right];$ (d) $\frac{d\phi }{d\tau }=-{10}^{10}{cosh}^{-2}\left[{10}^5{\varepsilon }^{-2}\tau \right]$.

Figure 5

Propagation of kink soliton in the network at frequency $f=777.275$ kHz for the above line parameters and the solution parameters $a_0=12, {\phi }_0=0.1, K_0=-5, K=0.1,$ and $\varepsilon ={10}^{-2}$. (a) Evolution of kink soliton at three different cells; (b) evolution of kink soliton at cell $n=25$ for three values of the dissipative parameter $\sigma$; (c) profile of the kink soliton at three different times.

Figure 6

Effects of the dissipative element of the network on the kink soliton propagating through cell $n=15$ at frequency $f=777.275$ kHz.

Figure 7

Propagation of dark soliton in the network at frequency $f=935.515$ kHz with the network parameters shown in the text for the solution parameters $\varepsilon ={10}^{-3}, K=-0.1, K_0=500,$ and different functional parameter $\phi \left(\tau \right)$ defined from (a) $\frac{d\phi \left(\tau \right)}{d\tau }={\mathit{PK}}^2+a_0$; (b) $\frac{d\phi \left(\tau \right)}{d\tau }={\mathit{PK}}^2+a_0\left(1+0.5cos\left[\frac{{10}^6\tau }{{\varepsilon }^2}\right]\right);$ (c) $\frac{d\phi \left(\tau \right)}{d\tau }={\mathit{PK}}^2+a_{0}exp\left[\frac{{10}^5}{{\varepsilon }^2}\tau \right]$; (d) $\frac{d\phi \left(\tau \right)}{d\tau }={\mathit{PK}}^2+a_{0}exp\left[-\frac{{10}^5}{{\varepsilon }^2}\tau \right],$ where $a_0=2000$.