Dynamics Of Modulated Waves 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.

Dynamics of modulated waves in the spring-block model of earthquake with time delay

Dynamics of modulated waves in the spring-block model of earthquake with time delay

Bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of DNA

In this work, we investigate the bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of deoxyribonucleic acid (DNA). We start by showing that this model can adopt right-handed or left-handed behavior depending on the wave frequency that it propagates. Then, using the semi-discrete approximation, we show that the dynamics of modulated waves in the network is governed by a quintic nonlinear Schrödinger (QNLS). Applying the phase imprint technique, this QNLS equation is first reduced into a derivative QNLS, and then to a cubic–quintic Duffing oscillator equation. Based on the phase plane analysis, we present all phase portraits of the dynamical system. The obtained results show a number of new phase portraits which cannot exist without the phase imprint approach. The exact representations of the nonlinear localized waves corresponding to the homoclinic and heteroclinic orbits in the phase portrait of the dynamical system are given. These waves include bright soliton, kink and anti-kink solitons, dark soliton and grey solitons. In addition, the effects of phase imprint parameter on the wave-shape profile of these solutions are investigated. We find out that the phase imprint parameter considerably affects the amplitude and the width of each of the above enumerated solitary waves, while the phase imprint method leads to new type solutions.

Nonlinear wave transmission in a two-dimensional nonlinear electric transmission network with dissipative elements

A two-dimensional nonlinear discrete electric transmission network made of several well-known anharmonic modified Noguchi lines coupled transversely to one another with a linear inductor is considered and the dynamics of modulated waves are investigated. The linear analysis shows that increasing either the dispersive elements of the network, the coupling elements of the network, or the transverse wavenumber increases the propagating frequencies. Employing the reductive perturbation method, we establish that the dynamics of modulated waves in the network system are governed by a two-dimensional dissipative nonlinear Schrödinger (NLS) equation having, depending on both the longitudinal and transverse wavenumbers as well as on the network parameters, either the hyperbolic or the elliptic character. Through the derived dissipative NLS equation, the phenomenon of the baseband modulational instability (MI) of the electric system under consideration is investigated and the analytical expression for the baseband MI growth rate is presented. We show that the dispersive elements of the network system softens the baseband MI and enhances the baseband MI domain. Under the condition of the baseband MI, we find approximate modulated wave solutions of the governing equations which are then used for analytical investigation of the transmission of super rogue wave voltage, Peregrine soliton voltage, and bright solitary wave signals through the network system under consideration. The effects of the network and solution parameters such as the dispersive, coupling parameters, the carrier wavenumber, and the wavenumbers in both the longitudinal and transverse directions on the dissipative wave signals are presented.

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.

Engineering chirped Lambert W-kink signals in a nonlinear electrical transmission network with dissipative elements

The dynamics of modulated waves in a nonlinear discrete transmission network with dissipative elements are investigated analytically. The generalized cubic-quintic Ginzburg-Landau (GCQ-GL) equation with intrapulse Raman scattering (IRS) terms governing spatially damped slowly modulated wave propagation is derived. Considering modulated Stokes wave propagating in the network, we investigate their baseband modulational instability (MI) and show that there are significant changes for the bandwidth frequency where the network may exhibit baseband MI. We show that the MI growth rate increases with the propagating frequency. The analytical chirped Lambert W-kink soliton-like solution of the derived GCQ-GL equation is presented and used to analyze the dissipative effects on chirped Lambert W-kink signals propagating through our network system. We show that the amplitude decreases both in space and time, while the width and the velocity remains constant when the Lambert W-kink signal propagates along the dissipative network. The effects of the cubic-Raman contributions on (i) the propagating frequency, (ii) the baseband MI, as well as (iii) the dynamics of Lambert W-kink signals are also investigated. In particular, our results reveal that a decrease in the cubic-Raman contributions enhances the baseband MI. The results of this paper may be useful for theoretical study of the propagation of ultrashort pulses in a single-mode optical fiber, as well as for experimental realization of undistorted transmission of optical pulses in optical fibers and further understanding their optical transmission properties.

Bifurcation of gap solitary waves in a two-dimensional electrical network with nonlinear dispersion

Abstract In this paper, the behavior of gap solitary waves is investigated in a two-dimensional electrical line with nonlinear dispersion. Applying the semidiscrete approximation, we show that the dynamics of modulated wave in the network can be described by an extended nonlinear Schrodinger equation. With the aid of the dynamical systems approach, we examine the fixed points of our model equation and the bifurcations of its phase portrait are presented. Likewise, we derive the exact parametric representations of bright soliton, dark soliton, peak and anti-peak solitons, kink and anti-kink solitons, periodic solutions and some compacton solutions corresponding to the various phase portrait trajectories under different parameter conditions. We find out that the nonlinear dispersion considerably affects the dynamics of the system and leads to a number of new solitary-wave solutions, namely, peakon, periodic peakon, compacton solutions, which can not be observed when the dispersion is assumed linear.

Nonlinear continuous bi-inductance electrical line with dissipative elements: Dynamics of the low frequency modulated waves

The dynamics of modulated waves in a nonlinear bi-inductance transmission line with dissipative elements are examined. We show the existence of two frequency modes and carry out intensive investigations on the low frequency mode. Thanks to the multiple scales method, the behavior of these waves is investigated and the dissipative effects are analyzed. It appears that the dissipation coefficient increases with the carrier wave frequency. In the continuous approximation, we derive that the propagation of these waves is governed by the complex Ginzburg–Landau equation instead of the Korteweg–de-Vries equation as previously established. Asymptotic studies of the dynamics of plane waves in the line reveal the existence of three additional regions in the dispersion curve where the modulational phenomenon is observed. In the low frequency mode, we demonstrate that the network allows the propagation of dark and bright solitons. Numerical findings are in perfect agreement with the analytical predictions.

Dynamics of the plane and solitary waves in a Noguchi network: Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrödinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al. (Phys. Rev. E 91 022925 (2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.

Dynamics of modulated waves in a nonlinear microtubule RLC transmission line

We study the dynamics of modulated ionic waves in a microtubule modeled by a nonlinear RLC transmission line (Tuszynski et al., 2004; Sataric et al., 2009). We show through the application of a reductive perturbation technique that the network can be reduced in the continuum limit to the dissipative nonlinear Schrödinger equation. The processes of the modulational instability and the effects of the dissipative elements of the network on wave propagation are investigated. We perform numerical simulations which confirm analytical predictions and give rise to localized modulated wave formation. The results suggest that microtubules can be biological structures where short-duration nonlinear waves called electrical envelope solitons can be created and propagated.

Periodic soliton trains and informational code structures in an improved soliton model for biomembranes and nerves.

Many experiments have shown that the action potential propagating in a nerve fiber is an electromechanical density pulse. A mathematical model proposed by Heimburg and Jackson is an important step in explaining the propagation of electromechanical pulses in nerves. In this work, we consider the dynamics of modulated waves in an improved soliton model for nerve pulses. Application of the reductive perturbation method on the resulting generalized Boussinesq equation in the low-amplitude and weak damping limit yields a damped nonlinear Schrödinger equation that is shown to admit soliton trains. This solution contains an undershoot beneath the baseline ("hyperpolarization") and a "refractory period," i.e., a minimum distance between pulses, and therefore it represents typical nerve profiles. Likewise, the linear stability of wave trains is analyzed. It is shown that the amplitude of the fourth-order mixed dispersive term introduced here can be used to control the amount of information transmitted along the nerve fiber. The results from the linear stability analysis show that, in addition to the main periodic wave trains observed in most nerve experiments, five other localized background modes can copropagate along the nerve. These modes could eventually be responsible for various fundamental processes in the nerve, such as phase transitions and electrical and mechanical changes. Furthermore, analytical and numerical analyses show that increasing the fourth-order mixed dispersion coefficient improves the stability of the nerve signal.

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.

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.

Transverse Stability in the Discrete Inductance-Capacitance Electrical Network

This work investigates the dynamics of modulated waves in a coupled nonlinear LC transmission line. By means of a method based on the semi-discrete limit and in suitably scaled coordinates, we derive the two-dimensional NLS equation governing the propagation of slowly modulated waves in the network. The exact transverse solution is found and the analytical criteria of stability of this solution are derived. The condition for which the network can exhibit modulational instability is also determined. The exactness of this analytical analysis is confirmed by numerical simulations performed on the exact equation of the network.

Effect of Second-Neighbor Inductive Coupling on the Modulational Instability in a Coupled Line of Transmission

In this work, the dynamics of modulated waves in a couple nonlinear LC transmission line is investigated. By employing the standard linear stability analysis, the growth rate of the instability is derived as a function of the wave numbers and system parameters. The regions of modulational instability can then be obtained and the influence of second-neighbor occuring via inductors in a nonlinear transmission line can be studied. As result, we found that the second-neighbor couplings add new maxima of gain; increase the group velocity, and the magnitude of the wave. Therefore, the network becomes more stable to small external perturbations.

Dynamics of modulated waves in electrical lines with dissipative elements

By a means of a method based on the reductive perturbation method, we show that the amplitude of waves on the nonlinear electrical transmission lines (NLTLs) is described by the cubic-quintic complex Ginzburg-Landau (CGL) equation. Then, we revisit analytically and numerically the processes of modulational instability (MI). The evolution of dissipative modulated waves through the network is also examined, and we show that solitonlike excitations can be induced by MI. Analytical results, illustrating the nature of MI of plane-wave solution, are also found to be in good agreement with numerical findings.

Wave propagation of nonlinear modes and formation of bubble in a two-component helicoidal lattice

Through the multiple scale expansion, we show that the dynamics of modulated waves in a two-component helicoidal one-dimensional model is governed by the nonlinear Schrodinger (NLS) equation. We study the occurrence of patterns formation in such a system. The impact of the helicoidal coupling is also studied which reveals that high values of the helicoidal coupling constant contribute to regulate information and charge transport throughout the two-chain model. This is made possible due to the superposition of optical and acoustic modes. In addition, the model is used to investigate internal dynamics of DNA. It is shown that the opening of the double helix of the DNA itself is controlled by the resonance mode, where the unzipping of the double helix has been illustrated by numerical simulations. Furthermore, the shape of the eyelike shape previously observed in experiments of thermal denaturation of DNA is numerically obtained.

Higher Order Solitons in an Electrical Lattice

Dynamics of modulated waves in a nonlinear electrical bi-inductance line are theoretically examined by the means of a perturbation method. It has been recently established that in such lines, modul...

Soliton-like excitations in the one-dimensional electrical transmission line

The dynamics of modulated waves are studied in the one-dimensional discrete nonlinear electrical transmission line. The contribution of the linear dispersive capacitance is taken into account, and it is shown via the reductive perturbation method that the evolution of such waves in this system is governed by the higher-order nonlinear Schrodinger equation. Passing through the Stokes analysis, we establish a generalized criterion for the Benjamin-Feir instability in the network and determine the exact solutions of the obtained wave equation by using the Pathria-Morris approach.

Solitonlike excitations in a one-dimensional electrical transmission line

Dynamics of modulated waves is studied in a one-dimensional discrete nonlinear electrical transmission line. Contribution of a linear dispersive capacitance is appreciated and it is shown via a reductive perturbation method that evolution of such waves in this system is governed by a higher order nonlinear Schrödinger equation. Passing through the Stokes wave analysis, a generalized criterion for the Benjamin–Feir instability in the network is presented and exact solutions of the obtained wave equation are determined by the means of the Pathria and Morris approach.