_{1}

^{*}

We investigate the propagation of dark solitons in a nonlinear dissipative electrical line. We show that the dynamics of the line is reduced to an expanded Korteweg-de Vries-Burgers (KdVB) equation. By applying the perturbation theory to the KdVB equation, we obtain soliton-like pulse solutions. The numerical simulations of the discrete equation are carried out and show the possibility of the founding solution to spread through the line. The effect of the dissipation through soliton is also shown. A chaotic-like behavior can take place in the system during the propagation of dark solitons through the line.

In recent years, nonlinear discrete system has received considerable attention. There are two main reasons for this interest: the development of experimental techniques making it possible to realize experiment in complicated periodic nonlinear structure and the potential for all-optical switching applications. The nonlinear electrical lines are very convenient tools for studying quantitatively the fascinating properties of wave propagation in nonlinear dispersive media [_{1} and r_{2}, accounting for the transmission line losses, the linear inductance L, and the voltage-dependent capacitance C(V_{n}) are the circuit parameters. The resulting circuit is referred to as a distributed model of a nonlinear RLC transmission lines.

In this work, after writing the discrete equation of the NLTL trough the Kirchhoff’s laws, we apply the perturbative method to reduce the previous equation to the well-known KdV-Burgers equation. This equation has solitary wave solution namely the dark soliton. Then we numerically investigated the propagation of this dark soliton through the line. Numerical investigations show also the impact of the dissipation on the soliton and the possible chaotic behavior. The remainder of the paper is organized as follows: in Section 2 we present the model of the NLTL understudy and we derive the equation governing modulated waves in the network. In Section 3, numerical simulations confirm the propagation of solitary wave through the line. This paper is concluded in the last.

We consider a nonlinear transmission line which consists of a number of RLC blocks connected as illustrated in

where C_{1} is a constant corresponding to the capacitance of the nonlinear diode at the dcbias-voltage U_{0}. The nonlinear parameters _{n} stored in the n-th capacitor in the line, and U_{n} is the voltage across the n-th capacitor. For the diode the sign of

Next, measuring time in units of

Starting from the semi-discrete model given by Equation (3), we develop a continuum model in the standard way, i.e. the right-hand side of (3) can be approximated with partial derivatives with respect to distance_{n}_{ }_{+}_{ }_{1}(t) and U_{n}_{ }_{−}_{ }_{1}(t), both around the reference value U_{n}(t). We regard the following equation as a continuum model of the transmission line that retains the effect of discreteness in the fourth-order term:

Setting dimensionless voltage F, such that

with,

and a further transformation of constants and variables such as

where _{1} = r_{2} = 0) it can support the well-known pulse soliton solution of the KdV (Korteweg-de Vries) equation. The KdV equation is the generic model for the study of weakly nonlinear long waves. It arises in physical systems which involve a balance between nonlinearity and dispersion at leading-order. For example, it describes surface waves of long wave length and small amplitude on shallow water and internal waves in a shallow density-stratified fluid. Many other applications for the KdV equations so exist, such as plasma waves, Ross waves and magma flow. Also, the KdV equation is integrable. This means that a collision between KdV solitary waves is elastic; after the collision the solitons retain their original shape with the only memory of the collision being a phase shift. The explicit solution for interacting KdV solitons was developed using the bilinear transformation method, by Hirota [

In the case of finite

is obtained, where the quantity

where

where

Another particular case, when we set

In order to study the behavior of the soliton along the cascaded maps, we numerically solve Equation (3) using the fourth-order Runge-Kutta scheme with a normalized integration time step_{1} = 320 pF. At the input of the line, we launch as initial solution the dark solution obtained above.

It can be seen that the soliton keep its shape when traveling down the line. Dark soliton appears as an intensity dip in an infinitely extended constant background. It has been investigated in many theoretical and experimental papers [

systems. However, these states sometimes occur and propagate together in some systems.

In this paper, we have studied analytically and numerically dark soliton. These solutions have been also obtained in microtubules protein structure. We have used the nonlinear transmission line analog of this structure to derive a nonlinear lattice equation governing the voltage equation of the system. This lattice equation was then treated analytically, by means of the continuum medium approximation. The latter is a variant of the perturbation method, which takes into regard the discreteness of the system by considering the carrier (dark) wave as a discrete (continuum) object. This approach allowed us to derive, in the wavelength approximation, a nonlinear perturbed KdV equation. The KdV model was then used to predict the formation of dark soliton. Numerical simulations performed in the framework of the nonlinear lattice equation, with initial conditions borrowed from the exact KdV solution, revealed that the dark soliton propagated through the line. Furthermore, it was shown how certain physical parameters of the configuration may affect the results. We numerically found soliton profile and characteristics (amplitude and width) that are in very good agreement with the analytical predictions. Our study also shows the presence of chaos in the system.

S.Y.D. is very grateful for the support of the Higher Teachers’ Training College of Maroua through his Director, and to A. M and A. S.