New complex wave patterns to the electrical transmission line model arising in network system

1.   Introduction

In the modern century, almost all computational programs use energy including electrical transmission line (ETL) with voltage. Such a line is one of the most important properties of energy transfer concepts. It has been seen that these properties of lines may rotate the behaviors of electrons in ETL. In this regards, F. Kenmogne et al. ^[1] have investigated the ETL in compact-like pulse signals in terms of its stability equilibrium points and weak linear dispersions. Khan et al. ^[2] have observed heat dissipation in ETL circuit. They have introduced a new model to symbolize heat propagation in ETL which results in damages the electrical tools. Dynamic behaviors of the nonlinear models arising in ETL have been presented by Tian et al. in ^[3]. They have used the modified Zakharov-Kuznetsov equation to symbolize physical phenomena. With the help of several analytical tools, physical dynamical properties of ETL have been obtained. Motcheyo et al have studied on the Chameleon's behavior of modulable ETL in ^[4]. They have also derived a mathematical description of ETL, and also, investigated some numerical behaviors of the model. Yemele et al. ^[5] have modulated the mathematical structures which are dynamics of signals in the network with ETL by considering peaked wave propagation and gray compactons. Mostly, such applications have been observed in communication systems where solitons are used to codify data. They have obtained compact gray compacton and peakon structures. Kanaya et al. ^[6] have designed an electrical small planner antenna with ETL. This is important in matching circuit on the thin patterned circuit board. In this tiny device, they have utilized the concept between interdigital gap and ETL which is composed of coplanar waveguide. Kuusela and his team ^[7] have conducted an experiment on the original Toda lattice and the dissipative lattice via nonlinear ETL. This is realized in investigating of soliton phenomena in nonlinear discrete systems. In 1987, R.Uklejewski et al. ^[8] have analyzed the transmission of vibrations in a porous vibroisolator with ETL theory. Senel et al. ^[9] have investigated the correlation among electricity and economic simulations. They have characterized the porous damping element of a vibroisolator, and also, they have presented reflection of waves among filtration velocity and fluid pressure. E. Tala-Tebue et al. ^[10,11] have presented a nonlinear model defined by

where $\alpha, \beta, \varpi_{0}, \omega_{0}$ are real constants with non-zero or complex while $v = v(x, y, t)$ defines the voltage in lines. $\delta_{1}$ symbolize the space in longitudinal while $\delta_{2}$ is used to explain the space in the transverse direction. This nonlinear electrical transmission line model (NETLM) explains the wave distributions on the network lines ^[10,11]. Therefore, many new mathematical systems have been introduced to the literature and they have been investigated by various experts ^{[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,33,34,35]}.

The contents of this paper are as follows. Section 2 presents the sine-Gordon expansion method ^[29,30,31]. Section 3 presents some new mixed dark-bright optical soliton solutions to the Eq (1.1). The main conclusions are presented in the last section of the paper.

2.   The SGEM

In this section we discuss the general facts of SGEM. Consider the following sine-Gordon equation:

where $u = u(x, \; t)$ and $m$ is a real constant. Applying the wave transformation as $u = U(\xi)$, $\xi = \mu(x-ct)$ to Eq (2.1), yields the following nonlinear ordinary differential equation (NODE):

where $\mu$ is the amplitude of the travelling wave and $c$ is the velocity of the travelling wave. Reconsidering Eq (2.2), we can write its full simplification as:

where $K$ is the integration constant.

Substituting $K = 0$, $w(\xi) = \frac{U}{2}$ and $a^{2} = \frac{m^{2}}{\mu^{2}(1-c^{2})}$ in Eq (2.3), gives:

Putting $a = 1$, we have:

Equation (2.5) is variables separable equation, we obtain the following two significant equations from solving it;

where $p$ is the integral constant.

For the solution of the following nonlinear partial differential equation;

we consider the wave transformation as $u = U(\xi)$, $\xi = \mu(x-ct)$, which converting this equation following nonlinear ordinary differential equation (NODE)

In this NODE, according to the general properties of SGEM, it may be chosen that the trial solution form is

Equation (2.9) can be rewritten according to Eqs (2.6) and (2.7) as follows:

We determine the value $n$ under the terms of NODE by balance principle which is considered as a relationship between the highest degree of the nonlinear terms and highest order of nonlinear differential equation. Letting the coefficients of $\sin^{i}(w)\cos^{j}(w)$ to be all zero, yields a system of equations. Solving this system by using various computational programs gives the values of $A_{i}, \; B_{i}, \; \mu$ and $c$ which is being real or complex values. Obtaining the different values of these coefficients giving exact solutions to the considered model produce new physical important of nonlinear mathematical models. Finally, substituting the values of $A_{i}, \; B_{i}, \; \mu$ and $c$ in Eq (2.9), we obtain the new travelling wave solutions to Eq (2.8). The SGEM is an analytical method which is based on two properties of Sine-Gordon equation (SGE). SGE is very important in explaining the wave propagation of the mathematical model.

3.   Application of SGEM

In this sub-section, we apply SGEM to the Eq (1.1). With the help of travelling wave transform

Equation (1.1) may be converted the following nonlinear ordinary differential equation

Using balance principle, yields $n = 1$. Putting $n = 1$ into Eq (2.9) produces

and into Eq (2.10) gives

and

Inserting Eqs (3.4) and Eq (3.5) into Eq (3.2), gives an algebraic equation in trigonometric function including various form of $\sin^{i}(w)\cos^{j}(w)$. Getting the coefficients of trigonometric terms in the same power to zero, give a system. Solving this system with aid of symbolic software to obtain the values of the coefficients involved, we find following coefficients in each case obtained from the set of algebraic equation systems, and it gives the travelling wave solutions to Eq (1.1).

Case-1: If

produces following new mixed dark-bright optical soliton

where

Case-2: Choosing these coefficients as $A_{0} = \frac{\sqrt{c^{2}-\varpi^{2}_{0}\delta^{2}_{1}-\delta^{2}_{2}\omega^{2}_{0}}}{c\sqrt{2\beta}}$, $A_{1} = \frac{\sqrt{c^{2}-\varpi^{2}_{0}\delta^{2}_{1}-\delta^{2}_{2}\omega^{2}_{0}}}{c\sqrt{2\beta}}$, $B_{1} = i\frac{\sqrt{c^{2}-\varpi^{2}_{0}\delta^{2}_{1}-\delta^{2}_{2}\omega^{2}_{0}}}{c\sqrt{2\beta}}$, $k = \frac{2\sqrt{3}\sqrt{c^{2}-\varpi^{2}_{0}\delta^{2}_{1}-\delta^{2}_{2}\omega^{2}_{0}}}{\sqrt{\varpi^{2}_{0}\delta^{4}_{1}+\delta^{4}_{2}\omega^{2}_{0}}}$, $\alpha = \frac{3\sqrt{\beta}\sqrt{c^{2}-\varpi^{2}_{0}\delta^{2}_{1}-\delta^{2}_{2}\omega^{2}_{0}}}{c\sqrt{2}}$,

we can find other new mixed dark-bright optical soliton

where

Case-3: When $A_{0} = \frac{\alpha}{3\beta}$, $A_{1} = \frac{-\alpha}{3\beta}$, $B_{1} = \frac{-i\alpha}{3\beta}$, $k = \frac{-2\sqrt{6}i\sqrt{\alpha^{2}(\varpi^{2}_{0}\delta^{2}_{1}+\delta^{2}_{2}\omega^{2}_{0})}}{\sqrt{(2\alpha^{2}-9\beta)(\varpi^{2}_{0}\delta^{4}_{1}+\delta^{4}_{2}\omega^{2}_{0})}}$, $c = \frac{3i\sqrt{\beta (\varpi^{2}_{0}\delta^{2}_{1}+\delta^{2}_{2}\omega^{2}_{0})}}{\sqrt{2\alpha^{2}-9\beta}}$, produces another mixed soliton as

where

.

Case-4: If $A_{0} = \frac{\alpha}{3\beta}$, $A_{1} = 0$, $B_{1} = \frac{-\sqrt{2}\alpha}{3\beta}$, $\omega_{0} = \frac{-ik(2c^{2}\alpha^{2}-9\beta c^{2}+9\beta\delta^{2}_{2}\omega^{2}_{0})}{\sqrt{108c^{2}\alpha^{2}\beta+81k^{2}\beta^{2}\delta^{4}_{2}\omega^{2}_{0}}}$, $\delta_{1} = \frac{-\sqrt{12c^{2}\alpha^{2}+9k^{2}\beta\delta^{4}_{2}\omega^{2}_{0}}}{\sqrt{k^{2}(c^{2}(2\alpha^{2}-9\beta)+9\beta\delta^{2}_{2}\omega^{2}_{0})} }$, gives new bright optical soliton as

4.   Linear stability analysis

We consider the perturbed solution of the form

where the $a_1$ is a steady state of the solution of Eq (1.1). Putting Eq (4.1) into Eq (1.1), we get

Taking the linearization of Eq (4.2), we get

Letting that Eq (4.3) has solution of the form

where $k_{i}, i = 1, 2$ are the normalized wave number. Inserting Eq.(4.4) into Eq.(4.3), we get

Solving Eq (4.5) for $\Omega$, the result yields

or

From solution 1, if $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 > 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta > 0$ or $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 < 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta < 0$, then the real is always negative, in this case the dispersion is stable. If $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 < 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta > 0$ or $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 > 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta < 0$, then the real part is zero, so in this case, it is more difficult to assess the long term behavior in this case, and it is label as the marginally stable.

From solution 2, if $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 > 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta > 0$ or $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 < 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta < 0$, then the real is always positive, in this case the dispersion is unstable. If $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 < 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta > 0$ or $k_1^2 \varpi_0^2 \delta_1^2 (-12 + k_1^2 \delta_1^2) + k_2^2 \delta_2^2(-12 + k_2^2 \delta_2^2)\omega_0^2 > 0$ and $3 - 6 a_1 \alpha + 9 a_1^2 \beta < 0$, then the real part is zero, so in this case, it is more difficult to assess the long term behavior in this case, and it is label as the marginally stable.

5.   Conclusions

In this research, the newly presented sine-Gordon equation method has been developed. The newly presented technique gives variety of wave solutions when tested on the nonlinear electrical transmission line model. Dark, mixed dark-bright optical, singular and mixed singular solitons solutions are successfully constructed. The conditions which guarantee the existence of the valid solutions to this model are given. The 2-, 3-dimensional, contour and density graphs to this model have been plotted to observe voltage behaviors on the electrical transmission line. It can be observed from Figures 1, 3, 5, 6, 8 that voltage is travelling wave propagations in the same electrical line. From Figure 2, 4, 7, 9, it can be inferred that electrical flow is stable and density between suitable places on this line. In this sense, linear stability analysis has been also investigated the strain conditions for the stability of Eq (1.1). After considering results obtained in this paper, it is estimated that these results have one of the most important properties of gravitational potential properties with dark and bright solutions ^[32]. The sine-Gordon equation method is an efficient and powerful mathematical approach which may be used in generating varieties of wave solutions to different kind of nonlinear wave equations.

Acknowledgments

Authors would like to express their thanks for the reviewers' valuable comments to improve this paper.

Conflict of interest

The Authors declare that there is no conflict of interest.