Driven localized excitations in the acoustic spectrum of small nonlinear macroscopic and microscopic lattices

Observation of Traveling Breathers and Their Scattering in a Two-Fluid System

The observation of traveling breathers (TBs) with large-amplitude oscillatory tails realizes an almost 50-year-old theoretical prediction [E. A. Kuznetsov and A. V. Mikhailov, Stability of stationary waves in nonlinear weakly dispersive media, Zh. Eksp. Teor. Fiz. 67, 1717 (1974) ZETFA70044-4510[E. A. Kuznetsov and A. V. MikhailovSov. Phys. JETP 40, 855 (1975)] SPHJAR0038-5646] and generalizes the notion of a breather. Two strongly nonlinear TB families are created in a core-annular flow by interacting a soliton and a nonlinear periodic (cnoidal) carrier. Bright and dark TBs are observed to move faster or slower, respectively, than the carrier while imparting a phase shift. Agreement with model equations is achieved. Scattering of the TBs is observed to be physically elastic. The observed TBs generalize to many continuum and discrete systems.

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.

Nondegenerate Parametric Resonance in Large Ensembles of Coupled Micromechanical Cantilevers with Varying Natural Frequencies

We investigate the collective dynamics and nondegenerate parametric resonance (NPR) of coplanar, interdigitated arrays of microcantilevers distinguished by their cantilevers having linearly expanding lengths and thus varying natural frequencies. Within a certain excitation frequency range, the resonators begin oscillating via NPR across the entire array consisting of 200 single-crystal silicon cantilevers. Tunable coupling generated from fringing electrostatic fields provides a mechanism to vary the scope of the NPR. Our experimental results are supported by a reduced-order model that reproduces the leading features of our data including the NPR band. The potential for tailoring the coupled response of suspended mechanical structures using NPR presents new possibilities in mass, force, and energy sensing applications, energy harvesting devices, and optomechanical systems.

Energy thresholds of discrete breathers in thermal equilibrium and relaxation processes

So far, only the energy thresholds of single discrete breathers in nonlinear Hamiltonian systems have been analytically obtained. In this work, the energy thresholds of discrete breathers in thermal equilibrium and the energy thresholds of long-lived discrete breathers which can remain after a long time relaxation are analytically estimated for nonlinear chains. These energy thresholds are size dependent. The energy thresholds of discrete breathers in thermal equilibrium are the same as the previous analytical results for single discrete breathers. The energy thresholds of long-lived discrete breathers in relaxation processes are different from the previous results for single discrete breathers but agree well with the published numerical results known to us. Because real systems are either in thermal equilibrium or in relaxation processes, the obtained results could be important for experimental detection of discrete breathers.

All-phononic amplification in coupled cantilever arrays based on gap soliton dynamics

We present a mechanism of amplification of phonons by phonons on the basis of nonlinear band-gap transmission phenomenon. As a concept the idea may be applied to the various number of systems; however we introduce the specific idea of creating an amplification scenario in the chain of coupled cantilever arrays. One chain is driven at the constant frequency located in the upper band of the ladder system, thus no wave enters the system. However the frequency is specifically chosen to be very close to the maximum value of the frequency corresponding to the dispersion relation of the system. An amplification scenario happens when a counter phase pulse of the same frequency with a small amplitude is introduced to the second chain. If both signals exceed a threshold amplitude for the band-gap transmission a large amplitude soliton enters the system-therefore we have an amplifier. Although the concept may be applied in a variety of contexts, all-optical or all-magnonic systems, we choose the system of coupled cantilever arrays and represent a clear example of the application of the presented conceptual idea. Logical operations is the other probable field, where such a mechanism could be used, which might significantly broaden the horizon of the considered applications of band-gap soliton dynamics.

Supertransmission channel for an intrinsic localized mode in a one-dimensional nonlinear physical lattice

It is well known that a moving intrinsic localized mode (ILM) in a nonlinear physical lattice looses energy because of the resonance between it and the underlying small amplitude plane wave spectrum. By exploring the Fourier transform (FT) properties of the nonlinear force of a running ILM in a driven and damped 1D nonlinear lattice, as described by a 2D wavenumber and frequency map, we quantify the magnitude of the resonance where the small amplitude normal mode dispersion curve and the FT amplitude components of the ILM intersect. We show that for a traveling ILM characterized by a specific frequency and wavenumber, either inside or outside the plane wave spectrum, and for situations where both onsite and intersite nonlinearity occur, either of the hard or soft type, the strength of this resonance depends on the specific mix of the two nonlinearities. Examples are presented demonstrating that by engineering this mix the resonance can be greatly reduced. The end result is a supertransmission channel for either a driven or undriven ILM in a nonintegrable, nonlinear yet physical lattice.

Observing (non)linear lattice dynamics in graphite by ultrafast Kikuchi diffraction

In materials, the nature of the strain-stress relationship, which is fundamental to their properties, is determined by both the linear and nonlinear elastic responses. Whereas the linear response can be measured by various techniques, the nonlinear behavior is nontrivial to probe and to reveal its nature. Here, we report the methodology of time-resolved Kikuchi diffraction for mapping the (non)linear elastic response of nanoscale graphite following an ultrafast, impulsive strain excitation. It is found that the longitudinal wave propagating along the c-axis exhibits echoes with a frequency of 9.1 GHz, which indicates the reflections of strain between the two surfaces of the material with a speed of ∼4 km/s. Because Kikuchi diffraction enables the probing of strain in the transverse direction, we also observed a higher-frequency mode at 75.5 GHz, which has a relatively long lifetime, on the order of milliseconds. The fluence dependence and the polarization properties of this nonlinear mode are entirely different from those of the linear, longitudinal mode, and here we suggest a localized breather motion in the a-b plane as the origin of the nonlinear shear dynamics. The approach presented in this contribution has the potential for a wide range of applications because most crystalline materials exhibit Kikuchi diffraction.

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.

Homoclinic orbits and chaos in a pair of parametrically driven coupled nonlinear resonators

We study the dynamics of a pair of parametrically driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by G. Kovačič and S. Wiggins [Physica D 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Šilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Šilnikov orbits are confirmed numerically.

Discrete solitons in electromechanical resonators

We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.

Bulk and surface magnetoinductive breathers in binary metamaterials

We investigate theoretically the existence of bulk and surface discrete breathers in a one-dimensional magnetic metamaterial comprised of a periodic binary array of split-ring resonators; the two types of resonators used have different resonant frequencies caused by unequal slit sizes. We use the rotating-wave approximation and construct several types of breather excitations both for the energy-conserving as well as dissipative-driven case; we corroborate these approximate results trough numerically exact computations. We demonstrate that discrete breathers can appear spontaneously in the dissipative-driven system as a result of a fundamental instability.

Intrinsic localized modes in parametrically driven arrays of nonlinear resonators

We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrödinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory.

Experimental and numerical exploration of intrinsic localized modes in an atomic lattice

This review focuses attention on the experimental studies of intrinsic localized modes (ILMs) produced in driven atomic lattices. Production methods involve the application of modulational instability under carefully controlled conditions. One experimental approach is to drive the atomic lattice far from equilibrium to produce ILMs, the second is to apply a driver of only modest strength but nearby in frequency to a plane wave mode so that a slow transformation from large amplitude standing waves to ILMs takes place. Since, in either case, the number of ILMs produced is small, the experimental observation tool appropriate for this task is four-wave mixing. This nonlinear detection technique makes use of the nonlinearity associated with an ILM to enhance its signal over that produced by the more numerous, but linear, spin waves. The final topic deals with numerical simulations of a nonlinear nanoscale atomic lattice where the new feature is running ILMs.

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.

Magnetoinductive breathers in metamaterials

The existence and stability of discrete breathers (DBs) in one- and two-dimensional magnetic metamaterials (MMs), which consist of periodic arrangements (arrays) of split-ring resonators (SRRs), are investigated numerically. We consider different configurations of the SRR arrays, which are related to the relative orientation of the SRRs in the MM, in both one and two spatial dimensions. In the latter case we also consider anisotropic MMs. Using standard numerical methods we construct several types of linearly stable breather excitation in both Hamiltonian and dissipative MMs (dissipative breathers). The study of stability in both cases is performed using standard Floquet analysis. In both cases we find that the increase of dimensionality from one to two spatial dimensions does not destroy the DBs, which may also exist in the case of moderate anisotropy (in two dimensions). In dissipative MMs, the dynamics is governed by a power balance between the mainly Ohmic dissipation and driving by an alternating magnetic field. In that case it is demonstrated that DB excitation locally alters the magnetic response of MMs from paramagnetic to diamagnetic. Moreover, when the frequency of the applied field approaches the SRR resonance frequency, the magnetic response of the MM in the region of the DB excitation may even become negative (extremely diamagnetic).