Discrete Electrical Lattice Research Articles

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Solitary wave solutions and investigation the effects of different wave velocities of the nonlinear modified Zakharov–Kuznetsov model for the wave propagation in nonlinear media

The modified Zakharov-Kuznetsov (mZK) model convey a significant role to analyze the inner mechanism of physical compound phenomenon in the field of two-dimensional discrete electrical lattice, the electrical waves in cold plasmas, plasma physics, nonlinear optics, wave behaviors of deep oceans, etc. In this study, numerous new, advanced, and more general exact solitary travelling wave solutions are explored to the early stated nonlinear model by the aid of newly developed generalized exponential rational function (GERF) method through the traveling wave transformation. The established solutions are in terms of trigonometric functions, rational functions, hyperbolic functions, exponential rational functions, and complex-soliton solutions. The results reveal that the free parameters significantly influence the existence of traveling waves, as well as nature, shape, and stability. The established soliton solutions demonstrate that the method is effective, compatible, scientifically efficient and easily applicable to identify a variety of wave structures of nonlinear evolution models. The mathematical simulations of the results were conducted by sketching 3D and 2D structures for different values of the associated parameters by the aid of the Wolfram Mathematica program.

Travelling waves in discrete electrical lattice with nonlinear symmetric capacitor

We study the propagation of voltage in a model of the conventional right-handed transmission line with a nonlinear symmetric capacitor. Applying the quasidiscrete approximation to the nonlinear voltage equation of the line, we derive a nonlinear Schrödinger equation and find the bright and dark solutions which are used as the initial condition for the integration of the nonlinear lattice model. The full integration of the lattice shows the propagation of the nonlinear voltage on the right-hand side and its robustness in time. The density energy of the lattice at each time has a node around which bright voltage is localized while in the case of dark voltage, it has a node where it seems to drop to zero at the soliton center. The plane phases allow proposing the characteristic of bright and dark solitons around the zero voltage.

A variety of new soliton structures and various dynamical behaviors of a discrete electrical lattice with nonlinear dispersion via variety of analytical architectures

Nonlinear electrical lattices are powerful experimental tool for studying nonlinear dispersive media and creating opportunities for the realistic modeling of electrical solitons. In this work, we adopt the generalized auxiliary equation methods, the first integral method, and the Möbius transformation approach to extract a variety of traveling and solitary wave solutions of the nonlinear Salerno equation describing the nonlinear discrete electrical lattice with nonlinear dispersion. The kink, antikink, dark, bright, peakons, antipeakons, and periodic wave solutions are all derived. The existence criteria of solitons are described. It has been shown that for the selective values of arbitrary constants in an auxiliary equation, the generalized auxiliary equation method II is reduced to the new ‐model expansion method as well as the new extended auxiliary equation method. The impact of free parameters on the obtained solutions is investigated and graphically depicted using physical descriptions. The acquired results are important for the validity of numerical and experimental results and further understanding of the wave propagation in the nonlinear discrete electrical lattice. In addition, the stability analysis of the obtained solitary wave solutions is studied. Furthermore, using the maple software, all derived solutions were checked by re‐entering them into the considered equation.

Solitary Wave Solutions to the Modified Zakharov–Kuznetsov and the (2 + 1)‐Dimensional Calogero–Bogoyavlenskii–Schiff Models in Mathematical Physics

The modified Zakharov–Kuznetsov (mZK) and the (2 + 1)‐dimensional Calogero–Bogoyavlenskii–Schiff (CBS) models convey a significant role to instruct the internal structure of tangible composite phenomena in the domain of two‐dimensional discrete electrical lattice, plasma physics, wave behaviors of deep oceans, nonlinear optics, etc. In this article, the dynamic, companionable, and further broad‐spectrum exact solitary solitons are extracted to the formerly stated nonlinear models by the aid of the recently enhanced auxiliary equation method through the traveling wave transformation. The implication of the soliton solutions attained with arbitrary constants can be substantial to interpret the involuted phenomena. The established soliton solutions show that the approach is broad‐spectrum, efficient, and algebraic computing friendly and it may be used to classify a variety of wave shapes. We analyze the achieved solitons by sketching figures for distinct values of the associated parameters by the aid of the Wolfram Mathematica program.

Dispersive soliton solutions for the Salerno equation for the nonlinear discrete electrical lattice in the forbidden bandgaps

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented.

Modulational instability and soliton propagation in an alternate right-handed and left-handed multi-coupled nonlinear dissipative transmission network

Abstract We investigate analytically the dynamics of modulated waves in an alternate right-handed and left-handed multi-coupled nonlinear dissipative discrete electrical lattice, made of several of the well-known modified Noguchi electrical transmission network that are transversely coupled to one another by a linear capacitor C 2 . The damped two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) governing slowly modulated wave propagation in this network system is derived. Considering modulated Stokes waves propagating in the network, we establish their criteria of instability, derive the analytical expression of the modulational instability (MI) growth rate, and show that the presence of the dispersive element C S softens the instability of the network system. Through the MI growth rate, we establish that the instability features of our network system show the famous donut Benjamin–Feir instability region. With the help of the linear dispersion law, we show that the behavior of our network model depends on that of the propagating frequency f p . It can adopt purely right-handed (corresponding to monotone increasing f p ), purely left-handed (associated to monotone decreasing f p ) or composite right-/left-handed (for composite monotone increasing and monotone decreasing f p ) behaviors without changing its structure. Also, the network system under consideration supports purely backward (when f p is positive), purely forward (when f p is negative) or composite backward/forward (when f p changes its sign) traveling waves. It appears that the network is right-handed (RH) and/or composite right-/left-handed (RH/LH) and supports backward traveling waves only for very low frequencies and becomes left-handed and supports only forward traveling waves for only high frequencies. Though the exact and approximative soliton-like solutions of the derived CGLE of the network, we investigate analytically RH, LH, as well as RH/LH behaviors of the system and show how to manipulate the transversal wave number to modify the behavior of the network, as well as the width and the motion of the bright solitary voltage signals in the network. We show that our network system supports RH and LH bright and kink soliton signals that simultaneously propagate at the same frequency; we also show that it supports backward and forward LH bright and kink soliton signals which simultaneously propagate at opposite frequency.

Rogue wave signals in a coupled anharmonic network: effects of the transverse direction

In this paper, we consider the dynamics of modulated waves in a nonlinear two-dimensional discrete electrical lattice made of several well-known anharmonic modified Noguchi lines coupled transversely to one another with a linear inductor. In the linear limit and depending on the chosen frequency domain, we demonstrate that the network behaves either as a right-handed line or as a left-handed line. We also establish that the band pass character of the modified Noguchi line increases the bandwidth of the allowed frequencies for which the system adopts the left-handed behavior. Based on the reductive perturbation method in the semi-discrete approximation, we derive a (2 + 1)-dimensional nonlinear Schrodinger (2D-NLS) equation governing the dynamics of the small amplitude signals in the system. The specific nature of this amplitude equation is examined, and it appears that its hyperbolic or elliptic character strongly depends on the coupling inductor $$L_{2}$$ . Moreover, using a simple transformation, we reduce the obtained equation to a (1 + 1)-dimensional nonlinear Schrodinger equation (1D-NLS). The exact first- and second-order rogue wave’s solutions of this 2D-NLS equation are derived in bandwidth frequencies where the network may exhibit modulational instability. The results of our investigations display number of interesting rogue wave profiles which have not been observed in the scientific literature. Furthermore, we examine the effects of both the coupling linear inductor $$L_{2}$$ and the nonlinear dispersion factor $$\eta$$ on the characteristics of the rogue waves and also on their propagation through the network. It appears that $$L_{2}$$ and $$\eta$$ can be used to squeeze or stretch the shape of the rogue waves signals traveling in the network. These numerical results are in perfect agreement with the analytical predictions.

On mathematical analysis of a discrete electrical lattice with nonlinear dispersion

In this paper, we consider the discrete electrical lattice with nonlinear dispersion described by Salerno equation, Fig. 1. Stability of equilibrium points, limit cycles and flip and Hopf bifurcations of the system are discussed. New exact solutions of a continuous approximation of the discrete system in the upper forbidden band gap are obtained by two methods, namely, [Formula: see text]-expansion function method and [Formula: see text] expansion method. Numerical simulation is used to follow the dynamics of the system and to investigate its physical properties.

On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering

In this research, analytical and numerical solutions are studied of a two–dimensional discrete electrical lattice, which is mathematically represented by the modified Zakharov–Kuznetsov equation. Moreover, the stability property of the obtained analytical solutions is investigated based on the Hamiltonian system, and then it is used to evaluate the initial and boundary conditions that are used in the numerical investigation. Many kinds of analytical solutions are obtained, such as complex, exponential, hyperbolic, and trigonometric function solutions. The functioning of both schemes of both techniques is tested and investigated.

Alternate backward and forward waves in a coupled nonlinear transmission line

In this work, we investigate backward and forward waves in a coupled nonlinear discrete electrical lattice. It is made of several of the well-known Noguchi electrical transmission line that are transversely coupled to one another by an inductor $$L_{2}$$. Based on the linear dispersion law, we show that the behavior of this model depends on the wave frequency that it propagates. It can adopt purely right-handed, purely left-handed or composite right-/left-handed behaviors without changing its structure. It appears that for fixed line’s parameters, the network is right-handed for low frequencies and becomes left-handed for high frequencies. It also appears that the increase of the coupling linear inductor induces a decrease of the width of the bandpass filter in the left-handed region while it increases its width in the right-handed region. By means of a method based on the semi-discrete limit and in suitably scaled coordinates, we derive a two-dimensional NLS equation governing the propagation of slowly modulated waves in the network. The exact transverse bright solitary solution is found. Using this solution, we investigate numerically both right-handed and left-handed behaviours of the system and show how to manipulate the coupling inductor to modify the width and the motion of the bright solitary voltage signals in the network.

New Optical Soliton Solutions of Nolinear Evolution Equation Describing Nonlinear Dispersion

In this work, we examine two algorithm schemes, namely, Kudryashov expansion and Auxiliary equation method for obtaining new optical soliton solutions of the discrete electrical lattice models in nonlinear scheme (Salerno equation). Our solutions obtained here are include the hyperbolic, rational, and trigonometric functions. Our two used methods are proved to be effective and powerful methods in obtaining the exact solutions of nonlinear evolution equations (NLEEs).

Existence and dynamics of solitary waves in a two-dimensional Noguchi nonlinear electrical network

In this work, we investigate solitary waves in a nonlinear two-dimensional discrete electrical lattice. It is made of several of the well-known Noguchi electrical transmission lines, that are transversely or longitudinally coupled to one another by an inductor ${L}_{2}$ or ${L}_{1}$ and a capacitor ${C}_{2}$ or ${C}_{1}$ mounted in parallel. The linear dispersion law of the network is given and the effects of the transverse coupling elements ${L}_{2}$ and ${C}_{2}$ on the allowed bandwidth frequencies are examined. Using the continuum limit approximation, we show that the dynamics of the small amplitude signals in the network can be governed by a (2+1)-dimensional generalized modified Zakharov-Kuznetsov equation. The fixed points of our model equation are examined and the bifurcations of its phase portrait are presented, as functions of the wave velocity of the signals that are to propagate in lattice. Likewise, we derive exact explicit solutions that are possible under different wave velocities and for physically realistic values of the network's parameters. These include pulse, kink, and anti-kink wave solutions and correspond to some special level curves of the first integral of the model equation. We find out that the transverse coupling parameters considerably affect the characteristics of the waves that are propagated throughout the system. Direct numerical simulations are also performed on the exact equations of the network and the results are in agreement with the analytical predictions.

Exact transverse solitary and periodic wave solutions in a coupled nonlinear inductor–capacitor network

Through two methods, we investigate the solitary and periodic wave solutions of the differential equation describing a nonlinear coupled two-dimensional discrete electrical lattice. The fixed points of our model equation are examined and the bifurcations of phase portraits of this equation for various values of the front wave velocity are presented. Using the sine-Gordon expansion method and classic integration, we obtain exact transverse solutions including breathers, bright solitons, and periodic solutions.

New soliton solutions for a discrete electrical lattice using the Jacobi elliptical function method

This paper presents many new solutions of a modified Zakharov–Kuznetsov equation obtained by using the Jacobi elliptical function method. This equation is shown to model a two dimensional discrete electrical lattice. The solutions reported herein are of varied types and include hyperbolic and trigonometric solutions, as well as kink and bell-shaped solitons. The comparison of our results to well-known ones is done. The method used here is very simple and concise and can be also applied to other nonlinear partial differential equations. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used to codify data.

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.

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.

Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method

In this paper, through the generalized Riccati equation mapping method, we investigate soliton solutions in the upper and lower forbidden band gab of the Salerno equation describing nonlinear discrete electrical lattice. As a result, we obtain various hyperbolic and trigonometric functions solutions and for some appropriated parameters we obtain exact solutions including kink, antikink, breathers, and dark and bright solitons. The obtained solutions are useful for the signal transmission through the electrical lattice.

Supratransmission induced by waves collisions in a discrete electrical lattice

We numerically performed a way to produce a supratransmission phenomenon in the Salerno equation describing the dynamics of modulated waves in a discrete nonlinear transmission lattice. For the natural supratransmission phenomenon, there exists a threshold of amplitude for which energy can flow in the line. We show that gap transmission is possible with driven amplitude below the threshold due to the collision of different plane waves coming from both edges of the line. One of the two plane waves has a frequency in the forbidden gap, and another has a frequency in the allowed phonon band. During collision, the wave in the allowed band is considered as a perturbation of the ones in the forbidden gap.

Supratransmission phenomenon in a discrete electrical lattice with nonlinear dispersion

We introduce the driven Salerno equation describing the dynamics of modulated waves in a discrete nonlinear electrical transmission lattice submitted to a periodic driving source with constant amplitude. This equation admits only in the upper forbidden band gap a new analytical expression of the supratransmission threshold, Bthr, which depends on the nonlinear dispersion coefficient, μ. Bthr increases (decreases) as μ decreases (increases) and for the limit case where μ vanishes, the value of Bthr is similar to that obtained in the cubic discrete nonlinear Schrödinger equation. Theoretical predictions of the supratrasmission phenomenon are confirmed by numerical simulations. We finally show that the driving amplitude must be slightly above the threshold to achieve a good supratransmission. Otherwise, the supratransmission takes place but the wave is destroyed.