Discrete breathers in a nonlinear electric line: modeling, computation, and experiment

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.

Propagating intrinsic localized mode in a cyclic, dissipative, self-dual one-dimensional nonlinear transmission line

A well-known feature of a propagating localized excitation in a discrete lattice is the generation of a backwave in the extended normal mode spectrum. To quantify the parameter-dependent amplitude of such a backwave, the properties of a running intrinsic localized mode (ILM) in electric, cyclic, dissipative, nonlinear 1D transmission lines, containing balanced nonlinear capacitive and inductive terms, are studied via simulations. Both balanced and unbalanced damping and driving conditions are treated. The introduction of a unit cell duplex driver, with a voltage source driving the nonlinear capacitor and a synchronized current source, the nonlinear inductor, provides an opportunity to design a cyclic, dissipative self-dual nonlinear transmission line. When the self-dual conditions are satisfied, the dynamical voltage and current equations of motion within a cell become the same, the strength of the fundamental, resonant coupling between the ILM and the lattice modes collapses, and the associated fundamental backwave is no longer observed.

Discrete breathers in a mechanical metamaterial

We consider a previously experimentally realized discrete model that describes a mechanical metamaterial consisting of a chain of pairs of rigid units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions with frequencies inside the gap between optical and acoustic dispersion bands. We compute numerically exact solutions of this type for several different parameter regimes and investigate their properties and stability. Our findings demonstrate that upon appropriate parameter tuning within experimentally tractable ranges, the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling and symmetry-breaking bifurcations, in addition to other mechanisms of stability change such as saddle-center and Hamiltonian Hopf bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system and paving the way for potential further experimental exploration of this rich nonlinear dynamical lattice setting.

Experimental investigation of supertransmission for an intrinsic localized mode in a cyclic nonlinear transmission line

In this experimental study of the nonlinear loss mechanism between traveling localized excitation and the underlying extended normal mode spectrum for a 1D lattice, three types of cyclic, electric, nonlinear transmission lines (NLTLs) are used. They are nonlinear capacitive, inductive, and capacitive+inductive NLTLs. To maintain a robust, steady-state traveling intrinsic localized mode (ILM), a traveling wave driver is used. The ILM loses energy because of a resonance between it and the extended NLTL modes. A wake field excitation is detected directly from ILM velocity experiments by the decrease in ILM speed and by the observation of the wake. Its properties are quantified via a two-dimensional Fourier map in the frequency-wavenumber domain, determined from the measured spatial-time voltage pattern. Simulations support and extend these experimental findings. We find for the capacitive+inductive NLTL configuration, when the two nonlinear terms are theoretically balanced, the wake excitation is calculated to become very small, giving rise to supertransmission over an extended driving frequency range.

Discrete vortices on spatially nonuniform two-dimensional electric networks

Two-dimensional arrays of nonlinear electric oscillators are considered theoretically where nearest neighbors are coupled by relatively small constant but nonequal capacitors. The dynamics is approximately reduced to a weakly dissipative defocusing discrete nonlinear Schrödinger equation with translationally noninvariant linear dispersive coefficients. Behavior of quantized discrete vortices in such systems is shown to depend strongly on the spatial profile of the internode coupling as well as on the ratio between time-increasing healing length and lattice spacings. In particular, vortex clusters can be stably trapped for some initial period of time by a circular barrier in the coupling profile, but then, due to gradual dissipative broadening of vortex cores, they lose stability and suddenly start to move.

Taming intrinsic localized modes in a DNA lattice with damping, external force, and inhomogeneity

The dynamics of DNA in the presence of uniform damping and periodic force is studied. The damped and driven Joyeux-Buyukdagli model is used to investigate the formation of intrinsic localized modes (ILMs). Branches of ILMs are identified as well as their orbital stabilities. A study of the effect of inhomogeneity introduced into the DNA lattice and its ability to control chaotic behavior is conducted. It is seen that a single defect in the chain can induce synchronized spatiotemporal patterns, despite the fact that the entire set of oscillators and the impurity are chaotic when uncoupled. It is also shown that the periodic excitation applied on a specific site can drive the whole lattice into chaotic or regular spatial and temporal patterns.

Experimental and numerical observation of dark and bright breathers in the band gap of a diatomic electrical lattice

We observe dark and bright intrinsic localized modes (ILMs), also known as discrete breathers, experimentally and numerically in a diatomic-like electrical lattice. The experimental generation of dark ILMs by driving a dissipative lattice with spatially homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the band gap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band gap, the dark ILMs persist, until the nonlinear localization pattern reverses and bright ILMs appear on top of the finite background. Deep into the band gap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.

Discrete breathers dynamic in a model for DNA chain with a finite stacking enthalpy

The nonlinear dynamics of a homogeneous DNA chain based on site-dependent finite stacking and pairing enthalpies is studied. A new variant of extended discrete nonlinear Schrödinger equation describing the dynamics of modulated wave is derived. The regions of discrete modulational instability of plane carrier waves are studied, and it appears that these zones depend strongly on the phonon frequency of Fourier's mode. The staggered/unstaggered discrete breather (SDB/USDB) is obtained straightforwardly without the staggering transformation, and it is demonstrated that SDBs are less unstable than USDB. The instability of discrete multi-humped SDB/USDB solution does not depend on the number of peaks of the discrete breather (DB). By using the concept of Peierls-Nabarro energy barrier, it appears that the low-frequency DBs are more mobile.

Discrete breathers in an electric lattice with an impurity: Birth, interaction, and death

We have simulated aspects of intrinsic localized modes or discrete breathers in a modulated lumped transmission line with nonlinear varactors and a defect unit cell. As the inductance or capacitance of this cell is increased, a transition from instability to stability takes place. Namely, there exist threshold values of the inductance or capacitance of a lattice impurity for a breather to be able to attach to. A resistive defect can also anchor a breather. Moreover, by either gradually lowering all the source resistances, or else increasing the modulation frequency, multiple secondary ILMs can be spontaneously generated at host sites (with only a single inductive or capacitive defect). Further, if two impurities are subcritically spaced (the separation increasing with the amplitude of the modulation voltage), a breather can pop up midway, with no breathers at the impurity sites themselves. Finally, an ILM can pull closer its neighbors on both sides, only to perish once these ILMs have gotten sufficiently close. To our knowledge, these effects have not been reported for any discrete nonlinear system.

Impulse-induced generation of stationary and moving discrete breathers in nonlinear oscillator networks

We study discrete breathers in prototypical nonlinear oscillator networks subjected to nonharmonic zero-mean periodic excitations. We show how the generation of stationary and moving discrete breathers are optimally controlled by solely varying the impulse transmitted by the periodic excitations, while keeping constant the excitation's amplitude and period. Our theoretical and numerical results show that the enhancer effect of increasing values of the excitation's impulse, in the sense of facilitating the generation of stationary and moving breathers, is due to a correlative increase of the breather's action and energy.

Mobility of discrete multibreathers in the exciton dynamics of the Davydov model with saturable nonlinearities

We show that the state of amide-I excitations in proteins is modeled by the discrete nonlinear Schrödinger equation with saturable nonlinearities. This is done by extending the Davydov model to take into account the competition between local compression and local dilatation of the lattice, thus leading to the interplay between self-focusing and defocusing saturable nonlinearities. Site-centered (sc) mode and/or bond-centered mode like discrete multihump soliton (DMHS) solutions are found numerically and their stability is analyzed. As a result, we obtained the existence and stability diagrams for all observed types of sc DMHS solutions. We also note that the stability of sc DMHS solutions depends not only on the value of the interpeak separation but also on the number of peaks, while their counterpart having at least one intersite soliton is instable. A study of mobility is achieved and it appears that, depending on the higher-order saturable nonlinearity, DMHS-like mechanism for vibrational energy transport along the protein chain is possible.

Nonlinear localized modes in two-dimensional electrical lattices

We report the observation of spontaneous localization of energy in two spatial dimensions in the context of nonlinear electrical lattices. Both stationary and moving self-localized modes were generated experimentally and theoretically in a family of two-dimensional square as well as honeycomb lattices composed of 6 × 6 elements. Specifically, we find regions in driver voltage and frequency where stationary discrete breathers, also known as intrinsic localized modes (ILMs), exist and are stable due to the interplay of damping and spatially homogeneous driving. By introducing additional capacitors into the unit cell, these lattices can controllably induce mobile discrete breathers. When more than one such ILMs are experimentally generated in the lattice, the interplay of nonlinearity, discreteness, and wave interactions generates a complex dynamics wherein the ILMs attempt to maintain a minimum distance between one another. Numerical simulations show good agreement with experimental results and confirm that these phenomena qualitatively carry over to larger lattice sizes.

Modulational instability of a modified Gross-Pitaevskii equation with higher-order nonlinearity

We consider the modulational instability (MI) of Bose-Einstein condensate (BEC) described by a modified Gross-Pitaevskii (GP) equation with higher-order nonlinearity both analytically and numerically. A new explicit time-dependent criterion for exciting the MI is obtained. It is shown that the higher-order term can either suppress or enhance the MI, which is interesting for control of the system instability. Importantly, we predict that with the help of the higher-order nonlinearity, the MI can also take place in a BEC with repulsively contact interactions. The analytical results are confirmed by direct numerical simulations.

Generation of localized modes in an electrical lattice using subharmonic driving

We show experimentally and numerically that an intrinsic localized mode (ILM) can be stably produced (and experimentally observed) via subharmonic, spatially homogeneous driving in the context of a nonlinear electrical lattice. The precise nonlinear spatial response of the system has been seen to depend on the relative location in frequency between the driver frequency, ω(d), and the bottom of the linear dispersion curve, ω(0). If ω(d)/2 lies just below ω(0), then a single ILM can be generated in a 32-node lattice, whereas, when ω(d)/2 lies within the dispersion band, a spatially extended waveform resembling a train of ILMs results. To our knowledge, and despite its apparently broad relevance, such an experimental observation of subharmonically driven ILMs has not been previously reported.