Nonlinear Electrical Transmission Line Research Articles

Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line

Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions (kink/antikink solitons, singular, periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.

Comment on “Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line” [Phys. Lett. A 373 (2009) 3801–3809

We revisit the derivation of the equation modeling envelope waves in a discrete nonlinear electrical transmission line (NLTL) considered a few years back in Physics Letters A 373 (2009) 3801–3809. Using a combination of rotating wave approximation and the Gardner–Morikawa transformation, we show that the modulated waves are described by a new type of extended nonlinear Schrödinger equation. In addition the expressions of several coefficients of this equation are found to be strongly different from those given earlier. As a consequence, key relationships between these coefficients that sustained the previous analysis are broken.

Soliton excitation in the pass band of the transmission line based on modulation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11174145 and 11334005) and the Research Foundation for Young Scientists of Anhui University of Technology (Grant No. QZ201318).

We numerically investigate the excitation of soliton waves in the nonlinear electrical transmission line formed by many cells. When the periodic driving voltage with frequency in the pass band closing to the cutoff frequency is applied to the endpoint of the whole line, the soliton wave can be generated. The numerical results show that the soliton wave generation mainly depends on the self modulation associated with the nonlinear effect. In this study, the lower subharmonic component is also observed in the frequency spectrum. To further understand this phenomenon, we study the dependence of the subharmonic power spectrum and frequency on the forcing amplitude and frequency numerically, and find that the subharmonic frequency increases with the gradual growth of the driving amplitude.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (III)

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions.

Analytical study of dynamics of matter-wave solitons in lossless nonlinear discrete bi-inductance transmission lines

We consider a lossless one-dimensional nonlinear discrete bi-inductance electrical transmission line made of N identical unit cells. When lattice effects are considered, we use the reductive perturbation method in the semidiscrete limit to show that the dynamics of modulated waves can be modeled by the classical nonlinear Schrödinger (CNLS) equation, which describes the modulational instability and the propagation of bright and dark solitons on a continuous-wave background. Our theoretical analysis based on the CNLS equation predicts either two or four frequency regions with different behavior concerning the modulational instability of a plane wave. With the help of the analytical solutions of the CNLS equation, we investigate analytically the effects of the linear capacitance CS on the dynamics of matter-wave solitons in the network. Our results reveal that the linear parameter CS can be used to manipulate the motion of bright, dark, and kink soliton in the network.

Bifurcations and Exact Solutions of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (II)

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (I)

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

Exact solutions of the nonlinear differential—difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G′/G)-expansion method

We investigated exact traveling soliton solutions for the nonlinear electrical transmission line. By applying a concise and straightforward method, the variable-coefficient discrete (G′/G)-expansion method, we solve the nonlinear differential—difference equations associated with the network. We obtain some exact traveling wave solutions which include hyperbolic function solution, trigonometric function solution, rational solutions with arbitrary function, bright as well as dark solutions.

Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G′/G)-expansion method including the generalized Riccati equation

Using the Jacobi elliptic functions and the alternative (G′/G-expansion method including the generalized Riccati equation, we derive exact soliton solutions for a discrete nonlinear electrical transmission line in (2+1) dimension. More precisely, these methods are general as they lead us to diverse solutions that have not been previously obtained for the nonlinear electrical transmission lines. This study seeks to show that it is not often necessary to transform the equation of the network into a well-known differential equation before finding its solutions. The solutions obtained by the current methods are generalized periodic solutions of nonlinear equations. The shape of solutions can be well controlled by adjusting the parameters of the network. These exact solutions may have significant applications in telecommunication systems where solitons are used to codify or for the transmission of data.

Modulational Instability in a Pair of Non-identical Coupled Nonlinear Electrical Transmission Lines

In this work, we investigate the dynamics of modulated waves non-identical coupled nonlinear transmission lines. Traditional methods for avoiding mode mixing in identical coupled nonlinear electrical lines consist of adding the same number of linear inductors in each branch. Adding linear inductors in a single line leads to asymmetric coupled nonlinear electrical transmission lines which propagate the signal and the mode mixing. On one hand, the difference between the two lines induced the fission for only one mode of propagation. This fission is influenced by the amplitude of the signal and the amount of the input energy as well; it also narrows the width of the input pulse soliton, leading to a possible increasing of the bit rate. On the other hand, the dissymmetry of the two lines converts the network into a good amplifier for the ω_ mode which corresponds to the regime admitting low frequencies.

COMPACT-LIKE PULSE SIGNALS IN A NEW NONLINEAR ELECTRICAL TRANSMISSION LINE

A nonlinear electrical transmission line with an intersite circuit element acting as a nonlinear resistance is introduced and investigated. In the continuum limit, the dynamics of localized signals is described by a nonlinear evolution equation belonging to the family of nonlinear difiusive Burgers' equations. This equation admits compact pulse solutions and shares some symmetry properties with the Rosenau-Hyman K(2;2) equation. An exact discrete compactly- supported signal voltage is found for the network and the dissipative efiects on the pulse motion analytically studied. Numerical simulations conflrm the validity of analytical results and the robustness of these compact pulse signals which may have important applications in signal processing systems.

Experimental observation on asymmetric energy flux within the forbidden frequency band in the LC transmission line

We study the energy flux in a nonlinear electrical transmission line consisting of two coupled segments which are identical in structure and different in parameters. The asymmetry of energy flux caused by nonlinear wave has been observed experimentally in the forbidden band of the line. The experiment shows whether the energy can flow through the transmission line depends on the amplitude of the boundary driving voltages, which can be well explained in the theoretical framework of nonlinear supratransmission. The numerical simulation based on Kirchhoff’s laws further verifies the existence of the asymmetric energy flux in the forbidden band.

Dissipative Discrete System with Nearest-Neighbor Interaction for the Nonlinear Electrical Lattice

A generalized dissipative discrete complex Ginzburg-Landau equation that governs the wave propagation in dissipative discrete nonlinear electrical transmission line with negative nonlinear resistance is derived. This equation presents arbitrarily nearest-neighbor nonlinearities. We analyze the properties of such model both in connection to their modulational stability, as well as in regard to the generation of intrinsic localized modes. We present a generalized discrete Lange-Newell criterion. Numerical simulations are performed and we show that discrete breathers are generated through modulational instability.

Energy unidirectional transmission in an asymmetrically finite transmission line

The phenomenon of energy unidirectional transmission is numerically investigated by using a system of two coupled discrete nonlinear electrical transmission lines, each line of the network contains a finite number of cells and has different pass band structures, respectively. Using numerical simulations, we examine the frequency multiplication of the driving frequency and the lattice filtering effect in the line. These lead to the generation of energy unidirectional transmission. In the present work, energy is carried by the second harmonic wave in the pass band. In addition, we also study the dependence of the energy efficiency on the driving amplitude and other parameters of the model, such as the system size and the nonlinear coefficient, by calculation. Furthermore, after detailed numerical simulation, an experimental demonstration is realized. The experimental results agree with those in simulation qualitatively.

Exotic modulated signals in a nonlinear electrical transmission line: Modulated peak solitary wave and gray compacton

The dynamics of modulated signals in a nonlinear discrete electrical transmission line with the intersite nonlinearity is interpreted in terms of the extended nonlinear Schrödinger-type (ENLS) equation with nonlinear dispersion introduced by Yemele and Kenmogne (2009) [10]. We show that this ENLS equation may simplify as iA t + PA xx + Q∣ A∣ 2 A = ir 1∣ A∣ 2 A x + r 23 A ∗( A 2) xx + r 3 A(∣ A 2∣) xx and exhibits two branches of non smooth solutions according to the sign of the quantity μ 2 = ( 16 r 23 Q - r 1 2 ) / [ 64 r 23 ( r 23 + r 3 ) ] : a branch which contains a peak solitary wave when μ 2 > 0 and another containing gray compacton for μ 2 < 0. Exact analytical expressions for these solutions are derived as well as their properties, namely existence and stability. The exactness of this analytical analysis is confirmed by numerical simulations performed both on the ENLS equation and on the exact equations of the network. These solutions may have important applications in communication systems where solitons are used to codify data.

Asymmetric energy flux in a transmission line based on frequency multiplication

We experimentally investigate the energy flux in a nonlinear electrical transmission line formed by two segments which are identical in structure but have different parameters. By consideration of the driving voltage at a frequency within the passband of one segment, the phenomenon of asymmetric energy flux caused by linear waves has been observed. The experiment shows that the possibility of creation of an asymmetric energy flux mainly depends on the nonlinear and lattice effects; namely, the asymmetric energy flux here is carried by a linear wave in the passband, instead of by a nonlinear wave out of the band. To further understand this phenomenon in the transmission line, we study both experimentally and numerically the dependency of the asymmetric energy flux on the parameters, and the key factors which affect the efficiency of the asymmetric energy flux are obtained.

Nonlinear excitations in a continuous bi-inductance electrical line

The dynamics of nonlinear excitations in an electrical bi-inductance transmission line are examined by means of the multiple scales method. In the continuum approximation using an appropriate decoupling ansatz for the voltage of the two different cells, we consider modulated waves and show that their propagation in the network is governed by a nonlinear Schrödinger equation instead of a Korteweg–de Vries equation. We have also established the existence of two frequency modes and study separately the wave dynamics in each mode. It appears from our investigations that the continuous bi-inductance electrical line supports a dark soliton in the low frequency mode and only a bright soliton in the high frequency mode. The study of the network properties has revealed that a nonlinear bi-inductance electrical transmission line can be considered as a superposition of two independent nonlinear mono-inductance transmission lines with different inductances.

Exact traveling discrete kink-soliton solutions for the discrete nonlinear electrical transmission lines

We investigate exact soliton solutions for the discrete nonlinear electrical transmission line by performing the simplest equation method from a trivial seed solution. Starting from the nonlinear propagation of signals in electrical transmission lines, we derive exact traveling kink and antikink solitary wave solutions. It is shown that under a safe range of parameter, the shape of kink soliton can be controlled well by adjusting the parameter of the line. The analytical solutions for the kink and antikink solitary waves are tested in direct simulations.

Solitonic Ionic Currents Along Microtubules

Microtubules are very important cytoskeletal structures implicated in different cellular activities. We should mention the cell division and traffic of organelles (mitochondria, vesicles and other cargos) by kinesin and dynein motor proteins. Microtubules are also playing important roles in higher neuronal functions including mechanisms of consciousness and memory. We here investigate the circumstances and the conditions enabling microtubules to act as nonlinear electrical transmission lines for ions flow along their cylinders. We established a model in which each tubulin dimmer protein is an electric element with a capacitive, resistive and negative incrementally resistive property due to polyelectrolyte nature of microtubules in cytosol. The particular attention was paid to the role of nano-pores existing between neighboring dimmers within a MT wall which exhibit properties like ionic channels. These nano-pores are candidates to explain some properties of microtubules resembling to unipolar transistors enabling the rectification and amplification of ionic currents.

Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line

We introduce an extended nonlinear Schrödinger (ENLS) equation describing the dynamics of modulated waves in a nonlinear discrete electrical transmission line (NLTL) with nonlinear dispersion. We show that this equation admits envelope dark solitary wave with compact support, with width and speed independent of the amplitude, as a solution. Analytical criteria of existence and stability of this solution are derived. In particular, we show that the modulated compact wave may exist in the NLTL depending on the frequency range of the chosen carrier wave, for physically realistic parameters. The stability of compact dark solitary wave is confirmed by numerical simulations of this ENLS equation and the exact equations of the network.