Discrete Electrical Transmission Line Research Articles

Dynamic analysis of the effects of dissipative elements and modulational instability in a multicoupled nonlinear electrical transmission line with the propagation of new rogue waveforms

In this study, we consider a nonlinear multicoupled discrete electrical transmission line consisting of several modified Noguchi lines and analyze the dynamics of the effects of dissipative elements on modulated waves. This analysis shows that the dispersion element (C S ) and solution parameter (γ) strongly contribute to the increase in voltage amplitudes and to the modulation of these new rogue waveforms, unlike the dissipative element (G). Using a semi-discrete approximation, we demonstrate that the dynamics of modulated waves in such a dissipative electrical system can be governed by a system of nonlinear Schrödinger equations, the Manakov system, and system parameters. The phenomenon of modulational instability in this dissipative electrical system is studied, and areas of instability are shown. We found that the dissipative element of this system increased and decreased the areas of instability. Under the condition of this Manakov system, we determine the approximate modulated wave solutions that are then used for the dynamic analysis of the effects of dissipative elements when transmitting these new rogue waveforms through this dissipative electrical system. The effects of the parameters of this nonlinear dissipative electrical system, such as dispersive, dissipative, and solution parameters, in the dominant direction of propagation of these new rogue wave signals are presented. Based on these results, we observe that the effects of dissipative elements do exist in this nonlinear dissipative electrical system and that these dissipative elements would also impact the areas of modulational instability, which could gradually disappear in this electrical system.

Chirped solitons in discrete electrical transmission line

I¸n this paper, we investigate the solitary waves to the discrete nonlinear electrical transmission lines with resonant tunneling diode (RTD). We derive the envelope Ginzburg Landau (GL) equation to the lattice by employing reductive perturbation method in a small amplitude. We analyze the steady state of the results by using the modulation instability (MI). We also use the traveling wave transformation to investigate chirped bright and dark soliton to the GL equation. We demonstrate our results by some graphical representation in 3-D and 2-D.

Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear Schrödinger equation

The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrödinger (DNLS) equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated.

Hybrid dispersive media with controllable wave propagation: A new take on smart materials

In this paper, we report on the wave transmission characteristics of a hybrid one dimensional (1D) medium. The hybrid characteristic is the result of the coupling between a 1D mechanical waveguide in the form of an elastic beam, supporting the propagation of transverse waves and a discrete electrical transmission line, consisting of a series of inductors connected to ground through capacitors. The capacitors correspond to a periodic array of piezoelectric patches that are bonded to the beam and that couple the two waveguides. The coupling leads to a hybrid medium that is characterized by a coincidence condition for the frequency/wavenumber value corresponding to the intersection of the branches of the two waveguides. In the frequency range centered at coincidence, the hybrid medium features strong attenuation of wave motion as a result of the energy transfer towards the electrical transmission line. This energy transfer, and the ensuing attenuation of wave motion, is alike the one obtained through internal resonating units of the kind commonly used in metamaterials. However, the distinct shape of the dispersion curves suggests how this energy transfer is not the result of a resonance and is therefore fundamentally different. This paper presents the numerical investigation of the wave propagation in the considered media, it illustrates experimental evidence of wave transmission characteristics and compares the performance of the considered configuration with that of internal resonating metamaterials. In addition, the ability to conveniently tune the dispersion properties of the electrical transmission line is exploited to adapt the periodicity of the domain and to investigate diatomic periodic configurations that are characterized by a richer dispersion spectrum and broader bandwidth of wave attenuation at coincidence. The medium consisting of mechanical, piezoelectric, and analog electronic elements can be easily interfaced to digital devices to offer a novel approach to smart materials.

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.

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.

Multisolitons and stability of two hump solitons of upper cutoff mode in discrete electrical transmission line

We show that a discrete electrical transmission line, such as a Band-pass filter is modeled by the Salerno equation at the upper cutoff mode. Special interest is paid to the investigation of stationary localized solutions supported by this equation for some given experimental parameters. Applying a map approach, the profiles of single and two bright solitons are obtained. Linear stability and direct numerical simulations are performed and the results show that a single bright soliton is stable while two bright ones are unstable and lead to a single bright soliton. Finally, we show that the lifespan of two hump solitons increases depending on the length thrust range of the kicked initial condition.

Dispersive shock waves propagating in the cubic-quintic derivative nonlinear Schrödinger equation

The propagation of a dispersive shock wave is studied in a quintic-derivative nonlinear Schrödinger (Q-DNLS) equation, which may describe, for example, the wave propagation on a discrete electrical transmission line. It is shown that a physical system described by a Q-DNLS equation without a dissipative term may support the propagation of shock waves. The influence of the derivative nonlinearity terms on the shock is analyzed. Using the found exact shock solutions of the Q-DNLS equation as the initial input signal, we investigate numerically the spatiotemporal stability of the shock signal in the network.

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.

Patterns of traveling intrinsic localized modes in a driven electrical lattice

The emergence of very stable traveling intrinsic localized modes (ILMs) locked to a uniform driver is demonstrated in a discrete electrical transmission line. The speed of these traveling ILMs is tunable by the driver amplitude and frequency. It is found to be quite sensitive to the ratio of intersite to on-site nonlinearity. The number of traveling ILMs can also be selected via the driving conditions and appears to be the result of a spatiotemporal pattern selection process.

Experimental generation of intrinsic localized modes in a discrete electrical transmission line

Experimental evidence for the generation of intrinsic localized modes (ILMs) in a nonlinear electrical transmission line is presented both via modulational instability of the uniform mode and via driving the lattice locally. The spatial profiles of these modes localized on the order of the lattice spacing can be directly measured in this macroscopic lattice, in contrast to most other systems where ILMs have so far been detected.

Modulated waves and chaotic-like behaviours in the discrete electrical transmission line

The dynamics of a weakly nonlinear modulated wave in a discrete electrical transmission line with negative nonlinear resistance and dissipative effects is investigated. This leads to the propagation of envelope modulation in the line modelled by the discrete complex Ginzburg–Landau equation. For an initial modulated plane wave propagating in the line, the criteria for modulational instability as well as the threshold amplitude are derived. The evolution of modulated waves which have the shape of an envelope soliton and the chaotic-like behaviour of the system are also examined. We show that dissipation can affect patterns propagating into the system. Numerical simulations demonstrate the validity of theoretical predictions.

Nonlinear Schrödinger models and modulational instability in real electrical lattices

Abstract In nonlinear dispersive media, the propagation of modulated waves, such as envelope (bright) solitions or hole (dark) solitons, has been the subject of considerable interest for many years, as for example in nonlinear optics [A.C. Newell and J.V. Moloney, Nonlinear Optics (Addison-Wesley, 1991)]. On the other hand, discrete electrical transmission lines are very convenient tools to study the wave propagation in 1 D nonlinear dispersive media [A.C. Scott (Wiley-Interscience, 1970)]. In the present paper, we study the generation of nonlinear modulated waves in real electrical lattices. In the continuum limit, our theoretical analysis based on the Nonlinear Schrodinger equation (NLS) predicts three frequency regions with different behavior concerning the Modulational Instability of a plane wave. These predictions are confirmed by our experiments which show that between two modulationally stable frequency bands where hole solitons can be generated, there is a third band where spontaneous or induced modulational instability occurs and where envelope solitons exist. When lattice effects are considered the dynamics of modulated waves can be modeled by a discrete nonlinear Schrodinger equation which interpolates between the Ablowitz-Ladik and Discrete-self-trapping equations.