Cutoff solitons and bistability of the discrete inductance-capacitance electrical line: theory and experiments

Experimental study of intrinsic localized mode mobility in a cyclic, balanced, 1D nonlinear transmission line

Mobile intrinsic localized modes (ILMs) in balanced nonlinear capacitive-inductive cyclic transmission lines are studied by experiment, using a spatiotemporal driver under damped steady-state conditions. Without nonlinear balance, the experimentally observed resonance between the traveling ILM and normal modes of the nonlinear transmission line generates lattice drag via the production of a lattice backwave. In our experimental study of a balanced running ILM in a steady state, it is observed that the fundamental resonance can be removed over extended, well-defined driving frequency intervals and strongly suppressed over the complete ILM driving frequency range. Because both of these nonlinear capacitive and inductive elements display hysteresis our observation demonstrates that the experimental system, which is only partially self-dual, is surprisingly tolerant, regarding the precision necessary to eliminate the ILM backwave. It appears that simply balancing the cell dual nonlinearities makes the ILM envelope shape essentially the same at the two locations in the cell, so that the effective lattice discreteness seen by the ILM nearly vanishes.

Collective escape and supratransmission phenomena in a nonlinear oscillators chain

In this study, the collective escape and supratransmission phenomena along a nonlinear chain of coupled particles subjected to a cubic on-site potential are considered. It is shown that the minimum initial on-site amplitude for which there is a collective escape increases with the nonlinear coupling. When the chain is forced at one end by a periodical excitation, the system exhibits supratransmission phenomenon in both lower and upper forbidden bandgaps, and, subsequently, it appears that the driving amplitude threshold for supratransmission in the upper forbidden bandgap frequency decreases with the nonlinear coupling. Depending on the frequency range of the gap frequency, the collective escape and supratransmission can occur simultaneously; otherwise, the supratransmission prevails.

Propagation of modulated waves in narrow-bandpass one-dimensional lattices

We consider the problem of the propagation of modulated waves in one-dimensional discrete lattices with linear bandpass-type dispersion relation. We are interested specifically in the cases where the gap frequency f_{0} and the cutoff frequency f_{max} verify 0<f_{0}<f_{max}<2f_{0}. Analytical investigations of such models commonly use the rotating wave approximation in which only the fundamental harmonic is taken into consideration. This approach is routinely justified by the physical argument that the zeroth and higher harmonic frequency terms cannot propagate, being out of band. Using a simple model of electrical lattice as a case study, we show analytically that accounting for the generation of those terms is indispensable for the accurate prediction of modulated solitary waves supported by the model. Moreover, a carefully designed numerical investigation of the propagation of an exact and consistent second-order approximation of such waves reveals that the second-order harmonic included is transmitted throughout the network the same as the fundamental harmonic is. Thus, we unveil the pitfall of the rotating wave approximation that is detrimental to its reliability for most of the models to which it has commonly been employed. We also suggest how to avoid this pitfall.

Exact solitary wavelike solutions in a nonlinear microtubule RLC transmission line

Analytically, we study the dynamics of ionic waves in a microtubule modeled by a nonlinear resistor, inductor, and capacitor (RLC) transmission line. 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 are studied and, motivated with a solitary wave type of solution to the nonlinear Schrödinger (NLS) equation, we use the direct method and the Weierstrass's elliptic function method to present classes of solitary wavelike solutions to the dissipative NLS equation of the network. The results suggest that microtubules are the biological structures where short-duration nonlinear waves called electrical envelope solitons can be created and propagated. This work presents a good analytical approach of investigating the propagation of solitary waves through a microtubule modeled by a nonlinear RLC transmission line.

Waves transmission and amplification in an electrical model of microtubules

Inspired by standard electrophysiological models of microtubules, a discrete nonlinear equation for ionic wave propagation that incorporates a negative nonlinear resistance is presented. The conditions for wave propagation in forbidden band gap are analyzed without and with dissipation. The nonlinear response manifold method is used to determine the supratransmission threshold of the case of study without dissipation. This threshold is found to be similar to the value obtained by analytical methods. With the dissipation, the monitoring of the accumulated energy is used to estimate the infratransmission threshold. It appears that the value of the supratransmission threshold can be lower than the value of the infratransmission threshold. The system is found to amplify significantly the amplitude of the input signal, thus confirming known experimental results. Nevertheless, a proper choice of the parameter of the nonlinear resistance is required for further validation of our results. A possible biological implication of the obtained results is presented.

Experimental observation of the generation of cutoff solitons in a discrete LC nonlinear electrical line

We address the problem of supratransmission of waves in a discrete nonlinear system, driven at one end by a periodic excitation at a frequency lying above the phonon band edge. In an experimental electrical transmission line made of 200 inductance-capacitance LC cells, we establish the existence of a voltage threshold for a supratransmission enabling the generation and propagation of cut-off solitons within the line. The decisive role of modulational instability in the onset and development of the process of generation of cut-off solitons is clearly highlighted. The phenomenon of dissipation is identified as being particularly harmful for the soliton generation, but we show that its impact can be managed by a proper choice of the amplitude of the voltage excitation of the system.

Second-harmonic generation for dispersive elastic waves in a discrete granular chain

The propagation of nonlinear compressional waves in a one-dimensional granular chain driven at one end by a harmonic excitation is studied. The chain is described by a Fermi-Pasta-Ulam (FPU) lattice model with quadratic nonlinearity (α-FPU model), valid for strong initial compression of the chain by an external static force. A successive approximations method is used to obtain the analytical expressions for the amplitudes of the static displacement field and of the fundamental and second harmonics propagating through the lattice. Both propagating and evanescent second harmonics are shown to influence the nonlinear propagation characteristics of the fundamental frequency. The propagating regime is characterized by a periodic energy transfer between first and second harmonics, resulting from dispersion, which disappears when the second harmonic becomes evanescent.

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.

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.

Resonant emission of solitons from impurity-induced localized waves in nonlinear lattices

We propose a mechanism for soliton creation from resonantly excited localized waves via supratransmission in band gaps of nonlinear lattices. A nonlinear localized wave, which is formed by and vibrates around an impurity with an intrinsic frequency, is found to undergo a local resonance when subject to an external forcing. Under the resonance, an instability develops that leads to the efficient emission of solitons at a much lower rate than that in uniform lattices with no impurity.

Influence of impurities on solitons in the nonlinear LC transmission line

This paper studies the propagation of solitons in the nonlinear LC transmission line (NLCTL) with capacitor impurity. Based on Kirchhoff's laws, the numerical simulation shows that the amplitude of the soliton will be increased or decreased when it is close to the positive or negative impurity. Then, it will be split into reflected and transmitted waves by both the positive and negative impurities. Furthermore, their final amplitude and propagating speeds are almost independent of the impurity polarity. The observations near the impurity can be understood in the physical picture of the linear uncoupled energy absorption. By these results, we find that the impurity-soliton interactions (ISIs) in NLCTLs for both inductance and capacitance impurities, which have been seen to be different before, actually can be unified. They also indicate that the ISIs in NLCTLs essentially can be integrated with those in many other soliton systems, such as the Frenkel-Kontorova model and hydrodynamics. Moreover, the impurity-induced influence on the NLCTL solitons can also be well interpreted in the framework of the nonlinear Schrödinger model with an impurity term derived from the discrete voltage propagation equations by means of the perturbation method.

Localized short impulses in a nerve model with self-excitable membrane

During the generation and transmission of nerve impulses, the cytoplasm behaves like an excitable medium that self-regulates the shapes and magnitudes of the output excitation. In connection with this self-regulatory function, one can readily think of the plasma membrane as a nerve organ holding the key role in the mechanisms of generation and transmission of the transmembrane potential, namely, it is expected to provide the essential feedback that stabilizes the stimulus. Here, a simple and coherent picture of self-regulation of the nerve impulse is proposed in terms of one single feedback associated with the main excitable biological organ of the nervous system. In this purpose, an electrodynamic theory is developed within the framework of a cable model in which the membrane capacitor is regarded as a charge-management electrical component with a defined capacity-voltage characteristic. It is found that in both myelinated and myelin-free nerve fiber contexts, the transmembrane excitations are well-localized short impulses whose shape and stability are determined by the capacity-voltage characteristic assumed to govern the self-excitability properties of the nerve membrane.

Tristability in the pendula chain

Experiments on a chain of coupled pendula driven periodically at one end demonstrate the existence of a regime which produces an output frequency at an odd fraction of the driving frequency. The stationary state is then obtained with numerical simulations and modeled with an analytical solution of the continuous sine-Gordon equation that resembles a kinklike motion back and forth in the restricted geometry of the chain. This solution differs from the expressions used to understand nonlinear bistability where the synchronization constraint was the basic assumption. As a result the short pendula chain is shown to possess tristable stationary states and to act as a frequency divider.

Modulated waves and pattern formation in coupled discrete nonlinear LC transmission lines

The conditions for the propagation of modulated waves on a system of two coupled discrete nonlinear LC transmission lines with negative nonlinear resistance are examined, each line of the network containing a finite number of cells. Our theoretical analysis shows that the telegrapher equations of the electrical transmission line can be reduced to a set of two coupled discrete complex Ginzburg-Landau equations. Using the standard linear stability analysis, we derive the expression for the growth rate of instability as a function of the wave numbers and system parameters, then obtain regions of modulational instability. Using numerical simulations, we examine the long-time dynamics of modulated waves in the line. This leads to the generation of nonlinear modulated waves which have the shape of a soliton for the fast and low modes. The effects of dissipative elements on the propagation are also shown. The analytical results are corroborated by numerical simulations.