Nonlinear Electrical Transmission Line Research Articles

New exact traveling wave solutions of space-time fractional nonlinear electrical transmission lines equation: arising in electrical engineering

New exact traveling wave solutions of space-time fractional nonlinear electrical transmission lines equation: arising in electrical engineering

New Complex Solutions to the Nonlinear Electrical Transmission Line Model

AbstractIn this paper, with the help of an analytical approach, new complex singular and travelling dark solutionsto the nonlinear electrical transmission line are successfully constructed. 2D and 3Dfigures along with contour figures are plotted. Finally, at the end of manuscript, general conclusions about these novel findings, which differ from existing results, are given.

On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method

Abstract In this paper we investigate the solutions of the fractional partial differential-difference equations governing the dynamics of the voltage wave flowing inside two fractional (low pass and pass band) nonlinear electrical transmission lines (NETL). Through the discrete Tanh method, we derive the traveling wave solutions i.e (kink, dark, singular kink solitons) of two fractional partial differential-difference equations by using the fractional complex transform. The impact of the fractional order on the dynamical behavior of the analytical solutions in both models is highlighted. From our findings the perfect matching between the analytical and the numerical solutions is justified by the stability seen as the resulting wave propagates safely inside the fractional low pass NETL including real inductor. We get for the fractional pass band NETL a rise of nonlinear periodic waves and a train of multi periodic solitons during the numerical simulations. We show that, according to the modified Riemann–Liouville derivative properties, the findings can describe physical systems such as electrical systems and nonlinear electrical transmission lines with transient effect.

Interacting signal packets in a lossless nonlinear transmission network with linear dispersion

In the small amplitude limit, we use the reductive perturbation method and the continuum limit approximation to derive a coupled nonlinear Schrö dinger (CNLS) equation describing the dynamics of two interacting signal packets in a discrete nonlinear electrical transmission line (NLTL) with linear dispersion. With the help of the derived CNLS equations, we present and analyze explicit expressions for the instability growth rate of a purely growing modulational instability (MI). We establish that the phenomenon of the MI can be observed only for “small” nonzero modulation wavenumbers. Also, we point out the effects of the linear dispersive element, as well as of the frequencies of the signal packets, on the instability growth rate. It is shown that the linear dispersion and the frequencies of signal packets can be well used to control the instability domain. Through the CNLS equations, we analytically investigate the propagation of solitary waves in the network. Our analytical studies show four types of interaction of signal packets propagating in the network: bright–bright, dark–dark, bright–dark and dark–bright soliton interactions.

Two-Dimensional Brain Microtubule Structures Behave as Memristive Devices

Microtubules (MTs) are cytoskeletal structures that play a central role in a variety of cell functions including cell division and cargo transfer. MTs are also nonlinear electrical transmission lines that produce and conduct electrical oscillations elicited by changes in either electric field and/or ionic gradients. The oscillatory behavior of MTs requires a voltage-sensitive gating mechanism to enable the electrodiffusional ionic movement through the MT wall. Here we explored the electrical response of non-oscillating rat brain MT sheets to square voltage steps. To ascertain the nature of the possible gating mechanism, the electrical response of non-oscillating rat brain MT sheets (2D arrays of MTs) to square pulses was analyzed under voltage-clamping conditions. A complex voltage-dependent nonlinear charge movement was observed, which represented the summation of two events. The first contribution was a small, saturating, voltage-dependent capacitance with a maximum charge displacement in the range of 4 fC/μm2. A second, major contribution was a non-saturating voltage-dependent charge transfer, consistent with the properties of a multistep memristive device. The memristive capabilities of MTs could drive oscillatory behavior, and enable voltage-driven neuromorphic circuits and architectures within neurons.

Modulational instability and peak solitary wave in a discrete nonlinear electrical transmission line described by the modified extended nonlinear Schrödinger equation

The present work describes the propagation of plane and peak solitary waves in a modified extended nonlinear Schrodinger (MENLS) equation that was earlier shown to govern the dynamics of modulated waves in a discrete nonlinear electrical transmission line (DNLETL). Firstly, the analytic expression for the modulational instability gain is found and the influence of wavenumber and wave amplitude on the gain is derived. It is predicted that they can be used to control the occurrence of modulation instability phenomenon in the network. Afterwards, using the MENLS equation, we show that this model of nonlinear electrical transmission line admits peak solitary wave for physically realistic parameters of the system. Direct numerical simulations are performed on the exact equations of the lattice and the obtained results are in very good agreement with the analytical predictions.

Bundles of Brain Microtubules Generate Electrical Oscillations

Microtubules (MTs) are long cylindrical structures of the cytoskeleton that control cell division, intracellular transport, and the shape of cells. MTs also form bundles, which are particularly prominent in neurons, where they help define axons and dendrites. MTs are bio-electrochemical transistors that form nonlinear electrical transmission lines. However, the electrical properties of most MT structures remain largely unknown. Here we show that bundles of brain MTs spontaneously generate electrical oscillations and bursts of electrical activity similar to action potentials. Under intracellular-like conditions, voltage-clamped MT bundles displayed electrical oscillations with a prominent fundamental frequency at 39 Hz that progressed through various periodic regimes. The electrical oscillations represented, in average, a 258% change in the ionic conductance of the MT structures. Interestingly, voltage-clamped membrane-permeabilized neurites of cultured mouse hippocampal neurons were also capable of both, generating electrical oscillations, and conducting the electrical signals along the length of the structure. Our findings indicate that electrical oscillations are an intrinsic property of brain MT bundles, which may have important implications in the control of various neuronal functions, including the gating and regulation of cytoskeleton-regulated excitable ion channels and electrical activity that may aid and extend to higher brain functions such as memory and consciousness.

New Jacobi elliptic function solutions, solitons and other solutions for the (2 + 1)-dimensional nonlinear electrical transmission line equation

In this work, we use the new Jacobi elliptic function expansion method to find exact soliton solutions for a discrete nonlinear electrical transmission line in (2 + 1) dimension. Several new solutions have been obtained. The solutions found by the current method are of varied types and include hyperbolic and trigonometric solutions, as well as Jacobi elliptic solutions. We show that the existence of these solutions depends on the parameters of the network. Comparisons of our new results with the well-known results are obtained. The solutions found here may be also used in optical fibers to transport information. The method applied here is very simple and concise and can be also applied to other nonlinear partial differential equations.

On the analytical expression of the multicompacton and some exact compact solutions of a nonlinear diffusive Burgers’type equation

We consider the nonlinear diffusive Burgers’ equation as a model equation for signals propagation on the nonlinear electrical transmission line with intersite nonlinearities. By applying the extend sine-cosine method and using an appropriate modification of the Double-Exp function method, we successfully derived on one hand the exact analytical solutions of two types of solitary waves with strictly finite extension or compact support: kinks and pulses, and on the other hand the exact solution for two interacting pulse solitary waves with compact support. These analytical results indicate that the speed of the pulse compactons doesn’t depends explicitly on the pulse amplitude as has been expected for long, but rather on the dc-component associated to this trigonometric solution. More interesting, the interactions between the two pulse compactons induce only a phase shift even though they are close together. These analytical solutions are checked by means of numerical simulations.

New exact travelling wave solutions for space-time fractional nonlinear equations describing nonlinear transmission lines

Abstract In this paper, we examines the effectiveness of newly developed algorithms called the exp ( - ϕ ( ξ ) ) -expansion function method and generalized Kudryashov method for constructing new and important travelling wave solutions of space-time fractional nonlinear differential equations. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann Liouville derivative sense. The validity and reliability the proposed methods are tested by its applications it for describe the nonlinear low-pass electrical transmission lines and pulse narrowing nonlinear transmission lines respectively.

New exact solutions for a discrete electrical lattice using the analytical methods

This paper retrieves soliton solutions to an equation in nonlinear electrical transmission lines using the semi-inverse variational principle method (SIVPM), the $\exp(-\Omega(\xi))$ -expansion method (EEM) and the improved $\tan(\phi/2)$ -expansion method (ITEM), with the aid of the symbolic computation package Maple. As a result, the SIVPM, EEM and ITEM methods are successfully employed and some new exact solitary wave solutions are acquired in terms of kink-singular soliton solution, hyperbolic solution, trigonometric solution, dark and bright soliton solutions. All solutions have been verified back into their corresponding equations with the aid of the Maple package program. We depicted the physical explanation of the extracted solutions with the choice of different parameters by plotting some 2D and 3D illustrations. Finally, we show that the used methods are robust and more efficient than other methods. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used to codify data.

Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range ( $0<\alpha\le 1$ ) of the derivative operator and we found the traditional solutions for the limiting case of $\alpha =1$ . We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.

Chirped solitons in derivative nonlinear Schrödinger equation

The exact chirped solitons are derived from the derivative nonlinear Schrödinger equation (DNLS). The obtained chirps could help for either pulse compression or amplification in optical fiber and nonlinear electrical transmission line. The 22 new obtained chirped solitons and 22 soliton solutions of the DNLS could help in the understanding of the phenomena in which waves are governed by such equation.

Applications of IBSOM and ETEM for solving a discrete electrical lattice

This paper presents a number of new solutions obtained for solving the nonlinear electrical transmission lines by using the improved Bernoulli sub-ODE method and the extended trial equation method. The proposed solutions are kink soliton solution, hyperbolic solution, trigonometric solution and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and concise and can be also applied to other nonlinear partial differential equations. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to construct many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used for codify data.

Second neighbors inducing common frequencies for bright and dark solitons

In this work, the dynamics of modulated waves in a modified Noguchi nonlinear electrical transmission line is studied with the contribution of second neighbors. It comes from this analysis that the line is governed by a dissipative nonlinear Schrodinger equation. One observes that the second neighbors counterbalance the effect of the linear capacitor CS in the frequency domains. The second neighbors well influence the line by increasing its bandwidth, its group velocity and the magnitude of the wave during its propagation. In the dispersion curve, we show that there exits a new region for the modulational instability/stability compared to the work of Pelap et al. (Phys. Rev. E 91, 022925 (2015)). The exactness of the analytical studies is accredited by numerical calculations. The most important feature of the new region, i.e. the second neighbors, is that the same frequency allows the use of either a bright soliton or a dark soliton depending on the choice of an appropriated wavelength.

Two-dimensional solitary waves and periodic waves on coupled nonlinear electrical transmission lines

By using the dynamical system approach, the exact travelling wave solutions for a system of coupled nonlinear electrical transmission lines are studied. Based on this method, the bifurcations of phase portraits of a dynamical system are given. The two-dimensional solitary wave solutions and periodic wave solutions on coupled nonlinear transmission lines are obtained. With the aid of Maple, the numerical simulations are conducted for solitary wave solutions and periodic wave solutions to the model equation. The results presented in this paper improve upon previous studies.

Modulated compact-like pulse signals in a nonlinear electrical transmission line: A specific case studied

The nonlinear electrical transmission line (NLTL) with intersite nonlinearity, recently introduced by Kenmogne and Yemélé is reexplored, with a particular attention carried on their obtained extended nonlinear Schrödinger (ENLS) equation, that is the basic NLS equation with additional nonlinear dispersive terms. Through the dynamical studies and analysis of equilibrium points it is shown that in the absence of linear dispersion term, a new solution for the ENLS equation is found, that is the envelope bright solitary wave with compact support. Analytical criterion of existence of this compact bright enveloped soliton is derived, and its stability is confirmed through numerical simulations of the exact equations of the network. Finally, some potential applications in communication systems are suggested.

Chameleon’s behavior of modulable nonlinear electrical transmission line

We show that modulable discrete nonlinear transmission line can adopt Chameleon’s behavior due to the fact that, without changing its appearance structure, it can become alternatively purely right or left handed line which is different to the composite one. Using a quasidiscrete approximation, we derive a nonlinear Schrödinger equation, that predicts accurately the carrier frequency threshold from the linear analysis. It appears that the increasing of the linear capacitor in parallel in the series branch induced the selectivity of the filter in the right-handed region while it increases band pass filter in the left-handed region. Numerical simulations of the nonlinear model confirm the forward wave in the right handed line and the backward wave in the left handed one.

Traveling wave solutions along microtubules and in the Zhiber–Shabat equation

The aim of this paper is to apply the exponential rational function method for solving nonlinear equations arising in various physical models such as nonlinear electrical transmission lines, optical fibers, DNA, to mention a few. New exact solutions are obtained and can help to well understand the process of those systems. The method used is very effective and simple and can be applied to other types of nonlinear equations.

Comment on "Dynamics and properties of waves in a modified Noguchi electrical transmission line".

A recent paper [Phys. Rev. E 91, 022925 (2015)PRESCM1539-375510.1103/PhysRevE.91.022925] presents the derivation of the nonlinear equation modeling envelope waves in a specific case of band passed filter discrete nonlinear electrical transmission line (NLTL), called "A modified Noguchi electrical transmission line" according to the authors. Using the reductive perturbation approach in the semidiscrete approximation, they showed that the modulated waves propagating in this NLTL are described by the ordinary nonlinear Schrödinger (NLS) equation. On the basis of their results, the authors claimed that all previous works on the band passed filter NLTL, which considered the vanishing of the dc component of the signal voltage, are incorrect, and this dc term is nonzero. As a consequence, the dispersion and nonlinearity coefficients of the NLS equation are strongly different from those usually obtained, and they found, according to the sign of the product PQ, the existence of one more region (compared to the work of Marquié et al. [Phys.Rev.E 49, 828 (1994)]PLEEE81063-651X10.1103/PhysRevE.49.828) in the dispersion curve that allows the motion of envelope solitons of higher frequency in the system. In this Comment we provide sufficient theoretical and numerical evidence showing that the evidence obtained by the authors otherwise is due to certain terms missed in their mathematical developments when they derived the NLS equation. Our results also suggest that the previous work of Marquié and co-workers correctly predict the fact that the dc term of the signal voltage does not exist and there exist only two regions in the dispersion curve according to the sign of the product PQ.